Method of probe design for nucleic acid analysis by multiplexed hybridization

ABSTRACT

Disclosed is an analysis method useful in multiplexed hybridization-mediated analysis of polymorphisms, i.e., wherein a labeled nucleic acid of interest (“target”) interacts with two or more pairs of immobilized degenerate capture probes. In one embodiment, one member of each pair has a sequence that is complementary to the normal (“wild-type”) sequence in a designated location of the target, while the other member of each pair has a sequence that is complementary to an anticipated variant (“mutant” or “polymorph”) sequence in that location of the target. These methods permit selection of two or more probe pairs such that, for each pair of probes interacting with a given target strand, interaction of the target with a preferred member of the probe pair is optimized. Also interpreting results obtained by multiplexed hybridization of the target to two or more pairs of probes under conditions permitting competitive hybridization is disclosed.

RELATED APPLICATIONS

This application claims priority to U.S. Provisional Application No.60/468,839, filed May 7, 2003.

BACKGROUND

Recently, multiplexed methods have been developed suitable for genomeanalysis, detection of polymorphisms, molecular karyotyping, andgene-expression analysis, where the genetic material in a sample ishybridized with a set of anchored oligonucleotide probes. Understandingthe results, displayed when labeled subsequences in the target bind toparticular probes, is based on decoding the probe array to determine thesequence of the bound members.

Embodiments of such arrays include spatially encoded oligonucleotideprobe arrays, where the array is placed on a chip and particular probesare placed in designated regions of the chip. Also included are particleor microbead arrays, designated random encoded arrays, where theoligonucleotide probes are bound to encoded beads, and where theencoding identifies the probe bound thereto. See, e.g., U.S. applicationSer. No. 09/690,040 Such arrays allow one to conduct high-throughputmultiplexed hybridization involving thousands of probes on a surface.

One problem in multiplexed analysis is in interpreting the results. In aconventional multiplexed “reverse dot blot” hybridization assay, thetarget oligonucleotide strands are labeled, and the labeling isdetected, post-hybridization to the probe array, to ascertain bindingevents. The reliability of the final computational interpretation of thedata depends on understanding the errors due to unintended interactionsamong targets and probes, as probes and targets are multiplexed.

The standard approach to the thermodynamic analysis of nucleic acidhybridization invokes interactions between adjacent (“nearest-neighbor”)base-pairs to account for the thermodynamic stability of the duplexunder given experimental conditions including temperature, ionicstrength, and pH (see, e.g., Cantor & Smith, “Genomics”, WileyInterscience, 1999). A variety of commercial software packages for probeand primer design invoke this model to estimate the thermodynamicstability, or equivalently, the “melting temperature”, T_(m), usuallydefined as the temperature at which half of a given number of nucleicacid duplexes have separated into single strands—a phenomenon alsoreferred to as “denaturing” or “de-annealing.” The melting temperatureof a probe or primer—more precisely, the duplex formed by the probe orprimer and a complementary single oligonucleotide strand—represents thesingle most widely used parameter to guide the design of probes andprimers in assays involving hybridization. Many commercial softwarepackages are available for this purpose, e.g., OLIGO, VISUALOMP,PRIMERSELECT, ARRAY DESIGNER, PRIMER3, and others.

Duplex stability is significantly affected by the presence of mismatchedbases, so that even a single point mutation can substantially lower themelting temperature (FIG. 1). This phenomenon provides the basis for thestandard method of detecting such a mutant or polymorph in a designatedlocation within a target sequence using hybridization analysis: a pairof probes, one member complementary to the normal target composition atthe designated location, the other complementary to an anticipatedvariant, are permitted to interact with the target. For example, in a“reverse dot blot” or READ™ assay format, both probes are immobilizedand permitted to capture a portion of a labeled target molecule ofinterest that is placed in contact with the probes. It is the objectiveof such an analysis to optimize discrimination between the signalassociated with the “perfectly matched (PM)” probe having the sequencethat is perfectly matched to the target sequence and the signalassociated with the “mismatched (MM)” probe having the sequence thatdeviates from that of the target sequence in the designated location. Itis desirable for the signal intensity to indicate the amount of capturedtarget; an example would be target-associated fluorescence, where ahigher intensity signal indicates more capture. For a single pair ofPM/MM probes, optimal discrimination would be ensured by selecting thetemperature to fall between the respective melting temperatures, T_(m)^(PM) and T_(m) ^(MM)<T_(m) ^(PM) of perfectly matched and mismatchedprobes (FIG. 1). However, in multiplexed reactions, involving two ormore PM/MM probe pairs, no such choice of a single temperature generallywill be possible. In addition, multiplexed mutation analysis generallyhas to account for the possibility of cross-hybridization between otherthan cognate probes and targets, a possibility that exists wheneverprobes or targets display a sufficiently high degree of homology so asto interact with sufficient strength so as to stabilize a duplex undergiven experimental conditions.

Another source of error arises from competitive hybridization, where atarget consists of possibly two distinct subsequences mA and mB, whichcan be characterized by separately hybridizing the target with either amixture of specific probes mmA and control probes mmA, or a mixture ofspecific probes pmB and control probes mmB, respectively. In either casethe ratio of specific signal to the control signal, obtained from eachseparate experiment, indicates how often either message is present. Onthe other hand, contrary to one's expectations, if the two messages werequeried by ratios of the respective signals in a multiplexed experimentconsisting of all four probes pmA, mmA, pmB, and mmB, one finds theseratios to differ from their values in the earlier experiments and byamounts that cannot simply be explained by the statistical noise. Inparticular, if one of the ratio values decreases severely, the resultingfalse negative errors can become significant enough to affect themultiplexed assay results. The negative effects can worsen as the numberof multiplexed probes for a target increases.

There is a need for a method to correct for such errors, and also forassay design methods and probe selection, which will avoid or minimizesuch errors.

SUMMARY

The analysis method disclosed herein is useful in multiplexed mutationanalysis or, more generally, in hybridization-mediated multiplexedanalysis of polymorphisms under conditions permitting competitivehybridization, i.e., wherein single strands of a labeled nucleic acid ofinterest (“target”) interact with two or more pairs of immobilizeddegenerate capture probes. In one embodiment, one member of each pairhas a sequence that is complementary to the normal (“wild-type”)sequence in a designated location of the target, while the other memberof each pair has a sequence that is complementary to an anticipatedvariant (“mutant” or “polymorph”) sequence in that location of thetarget. The methods herein permit the selection of two or more probepairs such that, for each pair of probes interacting with a given targetstrand, the interaction of the target with a preferred member of theprobe pair is optimized.

Also disclosed is: (i) a method of interpreting results obtained bymultiplexed hybridization of the target to two or more pairs of probesunder conditions permitting competitive hybridization; and (ii)application of these methods of probe selection and interpretation tomultiplexed analysis of target strands involving the use ofenzyme-catalyzed elongation reactions such as the polymerase chainreaction. Preferably, in mutation analysis, the amount of duplex formedfor each probe is determined by recording an optical signature(“signal”) associated with the encoded bead displaying that probe suchthat the intensity of the optical signature is proportional to theamount of duplex formed. That is, probes are analyzed in pairs, with onemember complementary to the normal sequence and the other to thevariant, i.e., the mutant or polymorph. One method of analyzing therespective results is to determine the ratio of the signals generatedfrom mutant hybridizations and that from wild-type hybridizations, andto set relative ranges of values indicative of normal, heterozygousvariant, and homozygous variant.

The paired sequences often will differ by only one nucleotide, andtherefore, some degree of cross-hybridization is anticipated uponintroduction of normal and variant samples. This can reducediscrimination in signal, even where one is examining signal ratios.

The methods herein invoke a model of mass action in which a target, T,present at initial concentration, t, forms a hybridization complex(“duplex”), C, at concentration or lateral density, c, with immobilizedprobes, P, present at a concentration or lateral density, p, inaccordance with the relation c=Kpt in which K denotes an affinityconstant that sets the scale of the strength of interaction betweenprobe(s) and target(s). In a preferred embodiment, K represents asequence-specific quantity which is evaluated as described herein usingstandard methodology well-known in the art (see SantaLucia, PNAS USA 95,1460-1465 (1998)) by invoking certain thermodynamic parameters relatingto the strength of base-pairing interactions that mediate the formationof a duplex.

These methods also guide the selection of two or more probe pairs so asto maximize, for each pair of probes interacting with a given targetstrand, the ratio of a first intensity, I^(PM), indicating the amount ofduplex formed between target and perfectly matched (“PM”) probe and asecond intensity, I^(MM), indicating the amount of duplex formed betweentarget and mismatched (“MM”) probe in a pair of probes.

Also disclosed is a method for the correction of results followinghybridization of the target to sets of PM/MM pairs of probes (alsoherein meant to include sets of “degenerate” probes) under conditionspermitting competitive hybridization.

The probes and methods disclosed herein were extensively validated inmany patient samples, and demonstrated as capable of diagnosing CF.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a “melting” curve for a pair of PM and MM probes using thepercentage of hybridization complex (“duplex”) remaining as the orderparameter; a mismatch results in a reduction of the melting temperature.

FIG. 1A depicts a target T with a single region perfectly complementaryto probe P₁₁, and another region perfectly complementary to probe P₁₂.

FIG. 2 depicts competitive hybridization between multiple probes and asingle target.

FIG. 3 shows a state transition diagram for the possible interactions oftwo PM/MM probe pairs and a target, T, where the target exists in one of9 possible states: free target T, and bound target states TP_(ij) ^(k),where k=1, 2 is the binding site, j=1, 2 is the probe index, and i=1 formatched probe or 0 for mismatched probe. Cross-terms j≠k have loweraffinity constants.

FIG. 3A shows a state transition diagram for the possible interactionsof two PM/MM probe pairs and a target, T, under Model I, involving onlystates 1, 2, 3, 8, 9.

FIG. 3B shows a state transition diagram for the possible interactionsof two PM/MM probe pairs and a target, T, under Model II, involving onlystates 1, 4, 5, 6, 7.

FIG. 4 is the “Δ plot,” i.e., a universal representation of thediscrimination of signal achieved with each of four different probepairs each having different sequences, which separately formhybridization complexes with a target, as a function of the ratio of theinitial concentrations of target and probes.

FIG. 4A plots PM/MM ratios of hybridization complexes with a target forthe first pair of probes, as a function of the initial targetconcentration scaled by total (over all probes used under a given modelin the multiplexed reaction) initial probe concentration, obtained underModel I (only the first pair of probes) and under Full Model (in thepresence of the second pair of probes).

FIG. 4B plots PM/MM ratios of hybridization complexes with a target forthe second pair of probes, as a function of the initial targetconcentration scaled by total (over all probes used under a given modelin the multiplexed reaction) initial probe concentration, obtained underModel II (only the second pair of probes) and under Full Model (in thepresence of the first pair of probes).

FIG. 4C shows the intensities of the probe pair of Model I of FIG. 4A,as functions of the initial target concentration scaled by total initialprobe concentration.

FIG. 4D shows the intensities of the probe pair of Model II of FIG. 4B,as functions of the initial target concentration scaled by total initialprobe concentration.

FIGS. 4E and 4F show, in bar graph form, where the X coordinate is takenat 0.7 from the graphs of FIGS. 4C and 4D, respectively, the normalizedintensities for each probe pair A and B.

FIG. 4G shows, in bar graph form, the normalized intensities as in FIGS.4E and 4F in a multiplexed reaction where both probe pairs A and B arepresent.

FIGS. 5A and 5B show a map indicating shifts in the plots of interactionstrength between a target and a first PM/MM probe pair as a result ofthe presence of a second PM/MM probe pair

FIG. 6A shows sequences of PM and MM capture probes A327, B354, C381,D359, and E286, designed for the hybridization-mediated detection ofmutations in exon 11 of the CFTR gene.

FIG. 6B shows the affinity constants for the probes A327, D359, andC381.

FIG. 7 shows plots of the ratio of the normalized discrimination for afirst pair of PM/MM probes when allowed to hybridize alone, in thepresence of a second pair of PM/MM probes, and in the presence of bothsecond and third pairs of PM/MM probes forming hybridization complexeswith a target, as a function of the ratio of the initial concentrationsof target and all probes.

FIG. 8A shows the Δ-plots for probe C381, with probes A327 and D359added.

FIG. 8B shows the relative locations of the probes A327, C381 and D359on exon 11.

FIG. 8C shows the Δ-plots for probe A327, with probes C381 and D359added.

FIG. 8D shows that probe D359 shows no discrimination, as there is nomismatch D359 probe.

FIG. 9A is a table showing the possible target-probe interactions ofFIGS. 9B-9E.

FIGS. 9B to 9E show the results of wild type and mutant targets in thepresence of five capture probes to regions of the exon 11 of the cysticfibrosis transmembrane regulator (CFTR) gene, referred to as A327, B354,C381, D359, and E286.

FIGS. 10A and 10B show the results of combinations of probes, asindicated in the figures.

DETAILED DESCRIPTION Duplex Formation in Multiconstituent Reactions

To provide a basis for the quantitative description of competitivehybridization, one invokes a specific model of competitive hybridizationinvolving the interaction of two or more PM/MM pairs of probes with thesame target sequence. In such model, the array includes microbeads withprobes attached, and the bead sizes are relatively large compared to thesize of the probes. One assumes in this model that there are thousandsof copies of the same probe attached to a single bead, and that thebeads are spaced on a planar surface far enough apart in order to ensurethat a single target strand may only hybridize to probes on a singlebead. Thus, this assumption implies that the only possible complexesinvolve one target and one probe. The targets are obtained from a longerDNA, by PCR amplification with two primers to select clones of a regionthat are subjected to further characterization. Let T be a target with asingle region perfectly complementary to probe P₁₁ and another regionperfectly complementary to probe P₁₂ (see FIG. 1A). Let P₀₁ differ fromP₁₁ in one base (i.e., the Hamming distance between P₀₁ and P₁₁ equalsto 1, H(P₀₁; P₁₁)=1). If P₁₁ and P₀₁ are the only probes present, we canexpect that when we compare the concentration of the P₁₁ probes bound toT (denoted [TP₁₁]) to the concentration of the P₀₁ probes bound to T(denoted [TP₀₁]) the resulting ratio to be large, i.e.:

${\frac{\left\lbrack {TP}_{11} \right\rbrack}{\left\lbrack {TP}_{01} \right\rbrack}}1\text{,}$since their free energies are chosen to satisfy:ΔC(P ₀₁)<ΔG(P ₁₁)P₀₁ clearly “competes” with P₁₁ for the target T. Consider yet anotherprobe, P₀₂, that differs from P₁₁ in one base as well (H(P₁₁; P₀₂)=1),but at a location different from the one in P₀₁ (H(P₀₁; P₀₂)=2). ThenP₀₂ also competes with P₁₁, but not as much with P₀₁, since H(P₀₁;P₀₂)=2. Thus, in the presence of P₀₂, we expect

$\frac{\left\lbrack {TP}_{11} \right\rbrack}{\left\lbrack {TP}_{01} \right\rbrack}$to decrease, since [TP₀₁] does not decrease much, but [TP₁₁] does.However, in the presence of all four probes P₁₁, P₀₁, P₁₂, and P₀₂, theanalysis of the resulting “mutual competitions” poses a non-trivialproblem.

FIG. 2 depicts the competitive hybridization among probes for a target Twith two sites each of which is complementary (“perfectly matched”) to aparticular probe: P₁₁ and P₁₂, respectively, and also where mismatchedprobes are included (P₀₁ and P₀₂, respectively). FIG. 3 depicts thestate transition diagram for the interaction of the probes and thetarget in FIG. 2. For the sake of providing concrete predictions,nearest-neighbor (“NN”) model of thermodynamic duplex stability is usedto evaluate sequence-specific affinities, K, of probe-target complexes.

Nearest-Neighbor Model

The model of hybridization discussed so far treats the dynamics in termsof kinetic mass-action reactions and ignores both the mixing propertiesof the molecules and the exact physics of hybridization except forsimply acknowledging that the thermodynamics parameters depend onbase-pair composition. However, the process of hybridization actuallyinvolves the formation of base pairs between Watson-Crick-complementarybases. Namely, base pairing of two single stranded DNA molecules isdetermined by the fact that A (adenine) is complementary to T (thymine),and C (cytosine) is complementary to G (guanine). Such base pairing isdue to the formation of hydrogen bonds between the complementary bases;thus, this interaction is characterized primarily by the composition ofthe interacting strands. Another physical interaction, base stacking,characterizes the hybridization process, and it has been shown to dependon the sequence rather than the composition of the strands. As basestacking depends on short-range interactions, it is thought to beadequately described by the Nearest-Neighbor (NN) model.

In the NN model, it is assumed that the stability of a given base pairis determined by the identity and orientation of the neighboring basepairs. Thus, each thermodynamic parameter of the hybridization process,such as the change in enthalpy (ΔH), entropy (ΔS), and free energy (ΔG),is calculated as a sum of the contributions from each nearest-neighborpair along a strand, corrected by some symmetry and initiationparameters. As the enthalpy and entropy terms may be assumed to beindependent of temperature, they can be computed as follows:

${\Delta\; H} = {{\sum\limits_{x}{\Delta\; H_{x}}} + {\Delta\;{H({init})}} + {\Delta\;{H({sym})}}}$${\Delta\; S} = {{\sum\limits_{x}{\Delta\; S_{x}}} + {\Delta\;{S({init})}} + {\Delta\;{S({sym})}}}$where the terms ΔH_(x) and ΔS_(x) are tabulated for all ten possible NNdimer duplexes, as are the initiation and symmetry terms. The freeenergy computation is analogous:

${\Delta\; G} = {{\sum\limits_{x}{\Delta\; G_{x}}} + {\Delta\;{G({init})}} + {\Delta\;{G({sym})}}}$with the initiation and symmetry terms tabulated. The values ΔG_(x) forthe dimer duplexes have been tabulated previously and reported at 25° C.(see Breslauer, et al., “Predicting DNA Duplex Stability from the BaseSequence,” PNAS USA 83: 3746-3750, 1986) and at 37° C. (SantaLucia, J.Jr. “A Unified View of Polymer, Dumbbell, and Oligonucleotide DNANearest-neighbor Thermodynamics,” PNAS USA 95:1460-1465, 1998). Since ΔGdepends on the temperature, the values ΔG_(x) for the dimer duplexes canbe easily calculated from the corresponding ΔH_(x) and ΔS_(x) parametersby:ΔG _(x)(T)=ΔH _(x) −TΔS _(x)The ten distinct dimer duplexes arise as follows. Following the notationof Breslauer et al., supra, each dimer duplex is denoted with a“slash-sign” separating antiparallel strands, e.g., AG/TC denotes5′-AG-3′ Watson-Crick base-paired with 3′-TC-5′. Table 1 below lists allsixteen (=|{A, T, C, G}|²=4²) possible dimers, identifying theequivalent ones.

TABLE 1 $\begin{matrix}\frac{AA}{TT} & {\frac{AC}{TG} \equiv \frac{GT}{CA}} & {\frac{AG}{TC} \equiv \frac{CT}{GA}} & \frac{AT}{TA} \\\frac{CA}{GT} & {\frac{CC}{GG} \equiv \frac{GG}{CC}} & \frac{CG}{GC} & \frac{CT}{GA} \\\frac{GA}{CT} & \frac{GC}{CG} & \frac{GG}{CC} & \frac{GT}{CA} \\\frac{TA}{AT} & {\frac{TC}{AG} \equiv \frac{GA}{CT}} & {\frac{TG}{AC} \equiv \frac{CA}{GT}} & {\frac{TT}{AA} \equiv \frac{AA}{TT}}\end{matrix}\quad$

Since these simulations involve oligonucleotide probes, the parametersfor the initiation of duplex formation used drawn from the results inthe 1998 paper of SantaLucia, supra. There, two different initiationparameters were introduced to account for the differences betweenduplexes with terminal A-T and duplexes with terminal G-C. Theadditional “symmetry” parameter accounts for the maintenance of the C2symmetry of self-complementary duplexes. The table of parameters used inthese simulations, drawn from SantaLucia, supra, is duplicated in Table2 below for convenience.

TABLE 2 NN Parameters at 1M NaCl. ΔG is calculated at 37° C. ΔH ΔS ΔGInteraction kcal/mol cal/K · mol kcal/mol AA/TT −7.9 −22.2 −1.00 AT/TA−7.2 −20.4 −0.88 TA/AT −7.2 −21.3 −0.58 CA/GT −8.5 −22.7 −1.45 GT/CA−8.4 −22.4 −1.44 CT/GA −7.8 −21.0 −1.28 GA/CT −8.2 −22.2 −1.30 CG/GC−10.6 −27.2 −2.17 GC/CG −9.8 −24.4 −2.24 GG/CC −8.0 −19.9 −1.84 Init.w/term. G · C 0.1 −2.8 0.98 Init. w/term. A · T 2.3 4.1 1.03 Symmetrycorrection 0 −1.4 0.43The following example illustrates how the free energy can be computedaccording to the formula above, using the values from Table 2.

5′ C—G—A—A—G—T 3′ (SEQ ID NO.: 1)    * * * * * * 3′ G—C—T—T—C—A 5′ (SEQID NO.: 2)

$\begin{matrix}{{\Delta\; G} = {{\Delta\;{G\left( {{CG}/{GC}} \right)}} + {\Delta\;{G\left( {{GA}/{CT}} \right)}} + {\Delta\;{G\left( {{AA}/{TT}} \right)}} + {\Delta\;{G\left( {{AG}/{TC}} \right)}} +}} \\{{\Delta\;{G\left( {{GT}/{CA}} \right)}} + {\Delta\;{G\left( {{{{init}.\mspace{14mu} w}/G} \cdot C} \right)}} + {\Delta\;{G\left( {{{{init}.\mspace{14mu} w}/A} \cdot T} \right)}} + 0} \\{= {{- 2.17} - 1.30 - 1.00 - 1.28 - 1.44 + 0.98 + 1.03}} \\{= {{- 5.18}\mspace{14mu}{kcal}\text{/}{mol}}}\end{matrix}$

Since the duplex is not self-complementary, ΔG(sym)=0.

Affinity Constants

At equilibrium, the affinity constants K₁ ^(j) are given by the formula:K ₁ ^(j)=exp [−ΔG/RT],and ΔG due to stacking interactions is calculated as above. Other modelsmay be possible or desirable and may be readily incorporated as a moduleinto the computations performed by methods described herein.

A mathematical model to analyze the dynamics involved in a system likethe one above is described below. As before, the assumption is that thesteric effects prevent multiple probes from hybridizing to a singletarget strand (as probes are bound to large beads).

As shown in FIG. 3, one may observe a target strand T in one of thefollowing nine possible states:

(1) T (Target is unbound.)

(2) TP₁₁ ¹; (3) TP₀₁ ¹; (4) TP₁₂ ²; (5) TP₀₂ ²

(Target is bound by “specific” hybridization.)

(8) TP₁₁ ²; (9) TP₀₁ ²; (6) TP₁₂ ¹; (7) TP₀₂ ¹

(Target is bound by “non-specific” hybridization.)

Bound target states have form TP_(ij) ^(k), where jε{1,2} is the probeindex, and

$i = \left\{ \begin{matrix}1 & {{{for}\mspace{14mu}{matched}\mspace{14mu}{probe}},} \\0 & {{{for}\mspace{14mu}{mismatch}\mspace{14mu}{pobe}},}\end{matrix} \right.$and kε{1,2} is the binding site. States within each category arenumbered “left-to-right” with respect to location on the target.

The set of reversible reactions operating between unbound and boundstates can be written as shown below, where the forward and backwardreaction rates are indicated with k_(i,j) and k_(j,i), respectively.While the reaction rates themselves are difficult to compute, the ratios(affinity constants,

K_(i) ^(j)=k_(i,j)/k_(j,i)) may be computed from purely thermodynamicconsiderations, and are sufficient for the “equilibrium analysis.”

$\begin{matrix}{{T + P_{11}}\underset{k_{2,1}}{\overset{k_{1,2}}{\rightleftarrows}}{TP}_{11}^{1}} & {{T + P_{12}}\underset{k_{8,1}}{\overset{k_{1,8}}{\rightleftarrows}}{TP}_{12}^{1}} \\{{T + P_{01}}\underset{k_{3,1}}{\overset{k_{1,3}}{\rightleftarrows}}{TP}_{01}^{1}} & {{T + P_{02}}\underset{k_{7,1}}{\overset{k_{1,7}}{\rightleftarrows}}{TP}_{02}^{1}} \\{{T + P_{12}}\underset{k_{4,1}}{\overset{k_{1,4}}{\rightleftarrows}}{TP}_{12}^{2}} & {{T + P_{11}}\underset{k_{8,1}}{\overset{k_{1,8}}{\rightleftarrows}}{TP}_{11}^{2}} \\{{T + P_{02}}\underset{k_{6,1}}{\overset{k_{1,6}}{\rightleftarrows}}{TP}_{02}^{2}} & {{T + P_{01}}\underset{k_{9,1}}{\overset{k_{1,9}}{\rightleftarrows}}{TP}_{01}^{2}}\end{matrix}$To perform a stationary analysis, where these reactions are allowed torun to equilibrium, one begins by assuming that all complexes can bedistinguished. In such case, the ODE's (ordinary differential equations)describing the dynamics of the system as follows.Let

$\begin{matrix}{\overset{\_}{X} = \left( {X_{1},X_{2},X_{3},X_{4},X_{5},X_{6},X_{7},X_{8},X_{9}} \right)^{T}} \\{= \left( {\lbrack T\rbrack,\left\lbrack {TP}_{11}^{1} \right\rbrack,\left\lbrack {TP}_{01}^{1} \right\rbrack,\left\lbrack {TP}_{12}^{2} \right\rbrack,\left\lbrack {TP}_{02}^{2} \right\rbrack,} \right.} \\\left. {\left\lbrack {TP}_{12}^{1} \right\rbrack,\left\lbrack {TP}_{02}^{1} \right\rbrack,\left\lbrack {TP}_{11}^{2} \right\rbrack,\left\lbrack {TP}_{01}^{2} \right\rbrack} \right)^{T}\end{matrix}$Note that at equilibrium,

$\begin{matrix}{\frac{\mathbb{d}\overset{\_}{X}}{\mathbb{d}t} = {\overset{\_}{0}.\begin{matrix}{\frac{\mathbb{d}\lbrack T\rbrack}{\mathbb{d}t} = {{k_{2,1}\left\lbrack {TP}_{11}^{1} \right\rbrack} + {k_{3,1}\left\lbrack {TP}_{01}^{1} \right\rbrack} + {k_{4,1}\left\lbrack {TP}_{12}^{2} \right\rbrack} + {k_{5,1}\left\lbrack {TP}_{02}^{2} \right\rbrack} +}} \\{{k_{6,1}\left\lbrack {TP}_{12}^{1} \right\rbrack} + {k_{7,1}\left\lbrack {TP}_{02}^{1} \right\rbrack} + {k_{8,1}\left\lbrack {TP}_{11}^{2} \right\rbrack} + {k_{9,1}\left\{ {TP}_{01}^{2} \right\rbrack} -} \\{{{k_{1,2}\lbrack T\rbrack}\left\lbrack P_{11} \right\rbrack} - {{k_{1,3}\lbrack T\rbrack}\left\lbrack P_{01} \right\rbrack} - {{k_{1,4}\lbrack T\rbrack}\left\lbrack P_{12} \right\rbrack} - {{k_{1,5}\lbrack T\rbrack}\left\lbrack P_{02} \right\rbrack} -} \\{{{k_{1,6}\lbrack T\rbrack}\left\lbrack P_{12} \right\rbrack} - {{k_{1,7}\lbrack T\rbrack}\left\lbrack P_{02} \right\rbrack} - {{k_{1,8}\lbrack T\rbrack}\left\lbrack P_{11} \right\rbrack} - {k_{1,9}\left\lbrack P_{01} \right\rbrack}}\end{matrix}}} & (1) \\{\frac{\mathbb{d}\left\lbrack {TP}_{11}^{1} \right\rbrack}{\mathbb{d}t} = {{{k_{1,2}\lbrack T\rbrack}\left\lbrack P_{11} \right\rbrack} - {k_{2,1}\left\lbrack {TP}_{11}^{1} \right\rbrack}}} & (2) \\{\frac{\mathbb{d}\left\lbrack {TP}_{01}^{1} \right\rbrack}{\mathbb{d}t} = {{{k_{1,3}\lbrack T\rbrack}\left\lbrack P_{01} \right\rbrack} - {k_{3,1}\left\lbrack {TP}_{01}^{1} \right\rbrack}}} & (3) \\{\frac{\mathbb{d}\left\lbrack {TP}_{12}^{2} \right\rbrack}{\mathbb{d}t} = {{{k_{1,4}\lbrack T\rbrack}\left\lbrack P_{12} \right\rbrack} - {k_{4,1}\left\lbrack {TP}_{12}^{2} \right\rbrack}}} & (4) \\{\frac{\mathbb{d}\left\lbrack {TP}_{02}^{2} \right\rbrack}{\mathbb{d}t} = {{{k_{1,5}\lbrack T\rbrack}\left\lbrack P_{02} \right\rbrack} - {k_{5,1}\left\lbrack {TP}_{02}^{2} \right\rbrack}}} & (5) \\{\frac{\mathbb{d}\left\lbrack {TP}_{12}^{1} \right\rbrack}{\mathbb{d}t} = {{{k_{1,6}\lbrack T\rbrack}\left\lbrack P_{12} \right\rbrack} - {k_{6,1}\left\lbrack {TP}_{12}^{1} \right\rbrack}}} & (6) \\{\frac{\mathbb{d}\left\lbrack {TP}_{02}^{1} \right\rbrack}{\mathbb{d}t} = {{{k_{1,7}\lbrack T\rbrack}\left\lbrack P_{02} \right\rbrack} - {k_{7,1}\left\lbrack {TP}_{02}^{1} \right\rbrack}}} & (7) \\{\frac{\mathbb{d}\left\lbrack {TP}_{11}^{2} \right\rbrack}{\mathbb{d}t} = {{{k_{1,8}\lbrack T\rbrack}\left\lbrack P_{11} \right\rbrack} - {k_{8,1}\left\lbrack {TP}_{11}^{2} \right\rbrack}}} & (8) \\{\frac{\mathbb{d}\left\lbrack {TP}_{01}^{2} \right\rbrack}{\mathbb{d}t} = {{{k_{1,9}\lbrack T\rbrack}\left\lbrack P_{01} \right\rbrack} - {k_{9,1}\left\lbrack {TP}_{01}^{2} \right\rbrack}}} & (9) \\{{Let}\begin{matrix}{\overset{\_}{X} = \left( {X_{1},X_{2},X_{3},X_{4},X_{5},X_{6},X_{7},X_{8},X_{9}} \right)^{T}} \\{= \left( {\lbrack T\rbrack,\left\lbrack {TP}_{11}^{1} \right\rbrack,\left\lbrack {TP}_{01}^{1} \right\rbrack,\left\lbrack {TP}_{12}^{2} \right\rbrack,\left\lbrack {TP}_{02}^{2} \right\rbrack,} \right.} \\\left. {\left\lbrack {TP}_{12}^{1} \right\rbrack,\left\lbrack {TP}_{02}^{1} \right\rbrack,\left\lbrack {TP}_{11}^{2} \right\rbrack,\left\lbrack {TP}_{01}^{2} \right\rbrack} \right)^{T}\end{matrix}} & (10)\end{matrix}$Note that at equilibrium,

$\begin{matrix}{\frac{\mathbb{d}\overset{\_}{X}}{\mathbb{d}t} = {\overset{\_}{0}.}} & (11)\end{matrix}$Applying (11) to equations (2)-(9) yields

$\begin{matrix}{{{k_{1,2}\lbrack T\rbrack}\left\lbrack P_{11} \right\rbrack} = {{\left. {k_{2,1}\left\lbrack {TP}_{11}^{1} \right\rbrack}\Longrightarrow K_{1}^{2} \right. \equiv \frac{k_{1,2}}{k_{2,1}}} = \frac{\left\lbrack {TP}_{11}^{1} \right\rbrack}{\lbrack T\rbrack\left\lbrack P_{11} \right\rbrack}}} & (12) \\{{{k_{1,3}\lbrack T\rbrack}\left\lbrack P_{01} \right\rbrack} = {{\left. {k_{3,1}\left\lbrack {TP}_{01}^{1} \right\rbrack}\Longrightarrow K_{1}^{3} \right. \equiv \frac{k_{1,3}}{k_{3,1}}} = \frac{\lbrack T\rbrack\left\lbrack P_{01}^{1} \right\rbrack}{\lbrack T\rbrack\left\lbrack P_{01} \right\rbrack}}} & (13) \\{{{k_{1,4}\lbrack T\rbrack}\left\lbrack P_{12} \right\rbrack} = {{\left. {k_{4,1}\left\lbrack {TP}_{12}^{2} \right\rbrack}\Longrightarrow K_{1}^{4} \right. \equiv \frac{k_{1,4}}{k_{4,1}}} = \frac{\left\lbrack {TP}_{12}^{2} \right\rbrack}{\lbrack T\rbrack\left\lbrack P_{12} \right\rbrack}}} & (14) \\{{{k_{1,5}\lbrack T\rbrack}\left\lbrack P_{02} \right\rbrack} = {{\left. {k_{5,1}\left\lbrack {TP}_{02}^{2} \right\rbrack}\Longrightarrow K_{1}^{5} \right. \equiv \frac{k_{1,5}}{k_{5,1}}} = \frac{\left\lbrack {TP}_{02}^{2} \right\rbrack}{\lbrack T\rbrack\left\lbrack P_{02} \right\rbrack}}} & (15) \\{{{{k_{1,6}\lbrack T\rbrack}\left\lbrack P_{12} \right\rbrack} = {{\left. {k_{6,1}\left\lbrack {TP}_{12}^{1} \right\rbrack}\Longrightarrow K_{1}^{6} \right. \equiv \frac{k_{1,6}}{k_{6,1}}} = \frac{\left\lbrack {TP}_{12}^{1} \right\rbrack}{\lbrack T\rbrack\left\lbrack P_{12} \right\rbrack}}}{{{k_{1,7}\lbrack T\rbrack}\left\lbrack P_{02} \right\rbrack} = {{\left. {k_{7,1}\left\lbrack {TP}_{02}^{1} \right\rbrack}\Longrightarrow K_{1}^{7} \right. \equiv \frac{k_{1,7}}{k_{7,1}}} = \frac{\left\lbrack {TP}_{02}^{1} \right\rbrack}{\lbrack T\rbrack\left\lbrack P_{02} \right\rbrack}}}{{{k_{1,8}\lbrack T\rbrack}\left\lbrack P_{11} \right\rbrack} = {{\left. {k_{8,1}\left\lbrack {TP}_{11}^{2} \right\rbrack}\Longrightarrow K_{1}^{8} \right. \equiv \frac{k_{1,8}}{k_{8,1}}} = \frac{\left\lbrack {TP}_{11}^{2} \right\rbrack}{\lbrack T\rbrack\left\lbrack P_{11} \right\rbrack}}}{{{k_{1,9}\lbrack T\rbrack}\left\lbrack P_{01} \right\rbrack} = {{\left. {k_{9,1}\left\lbrack {TP}_{01}^{1} \right\rbrack}\Longrightarrow K_{1}^{9} \right. \equiv \frac{k_{1,9}}{k_{9,1}}} = \frac{\left\lbrack {TP}_{01}^{1} \right\rbrack}{\lbrack T\rbrack\left\lbrack P_{01} \right\rbrack}}}} & (16)\end{matrix}$and applying it to equation (1) yields

$\begin{matrix}{{{k_{2,1}\left\lbrack {TP}_{11}^{1} \right\rbrack} + {k_{3,1}\left\lbrack {TP}_{01}^{1} \right\rbrack} + {k_{4,1}\left\lbrack {TP}_{12}^{2} \right\rbrack} + {k_{5,1}\left\lbrack {TP}_{02}^{2} \right\rbrack} + {k_{6,1}\left\lbrack {TP}_{12}^{1} \right\rbrack} + {k_{7,1}\left\lbrack {TP}_{02}^{1} \right\rbrack} + {k_{8,1}\left\lbrack {TP}_{11}^{2} \right\rbrack} + {k_{9,1}\left\lbrack {TP}_{01}^{2} \right\rbrack}} = {\quad{\lbrack T\rbrack\left( {{k_{1,2}\left\lbrack P_{11} \right\rbrack} + {k_{1,3}\left\lbrack P_{01} \right\rbrack} + {k_{1,4}\left\lbrack P_{12} \right\rbrack} + {k_{1,5}\left\lbrack P_{02} \right\rbrack} + {k_{1,6}\left\lbrack P_{12} \right\rbrack} + {k_{1,7}\left\lbrack P_{02} \right\rbrack} + {k_{1,8}\left\lbrack P_{11} \right\rbrack} + {k_{1,9}\left\lbrack P_{01} \right\rbrack}} \right)}}} & (20)\end{matrix}$Equation (20) is a linear combination of (12), . . . , (19), and henceprovides no additional information. Observe that

$\begin{matrix}{\frac{\mathbb{d}\lbrack T\rbrack}{\mathbb{d}t} = {{- \frac{\mathbb{d}}{\mathbb{d}t}}\left\{ {\left\lbrack {TP}_{11}^{1} \right\rbrack + \left\lbrack {TP}_{01}^{1} \right\rbrack + \left\lbrack {TP}_{12}^{2} \right\rbrack + \left\lbrack {TP}_{02}^{2} \right\rbrack +} \right.}} \\{\left. {\left\lbrack {TP}_{12}^{1} \right\rbrack + \left\lbrack {TP}_{02}^{1} \right\rbrack + \left\lbrack {TP}_{11}^{2} \right\rbrack + \left\lbrack {TP}_{01}^{2} \right\rbrack} \right\}\mspace{20mu}}\end{matrix}$${{or}\;(1)} = {- {\sum\limits_{j = {(2)}}^{(9)}\left\{ {{equation}\mspace{14mu} j} \right\}}}$The constants K₁ ^(j) for jε{2, . . . , 9}, appearing in equations(12)-(19), can be computed from probe sequence data. For each j,ΔG _(total) =−RT ln K ₁ ^(j),where R is the gas constant and T is the temperature (in degreesKelvin). Thus, we have

$\begin{matrix}{{K_{1}^{j} = {{\exp\left\lbrack {{- \Delta}\;{G_{total}/{RT}}} \right\rbrack}\text{,}\mspace{14mu}{where}}}{{\Delta\; G_{total}} = {{- \left( {\underset{\underset{initiation}{︸}}{\Delta\; g_{i}} + \underset{\underset{symmetry}{︸}}{\Delta\; g_{symm}}} \right)} + {\sum\limits_{T}{\underset{\underset{{sequence}\mspace{14mu}{data}}{︸}}{\Delta\; g_{x}}.}}}}} & (21)\end{matrix}$This notation and form follows the calculations set forth in Breslaueret al., supra; however, the calculation of ΔG has since been modified tofollow the methods later-disclosed in SantaLucia, supra. Both versionsof calculating ΔG are suitable for the model described herein.

The described model can be used to predict equilibrium concentrations ofcomplexes TP_(ij){iε{0, 1}, jε{1, 2}}, where K₁ ^(j) can be calculatedfrom the equation above (21), where ΔG_(total) is computed based on thesequence information. The following conservation rules must hold:

$\begin{matrix}{\left\lbrack P_{11} \right\rbrack_{0} = {\left\lbrack P_{11} \right\rbrack + \left\lbrack {TP}_{11}^{1} \right\rbrack + \left\lbrack {TP}_{11}^{2} \right\rbrack}} & (22) \\{\left\lbrack P_{01} \right\rbrack_{0} = {\left\lbrack P_{01} \right\rbrack + \left\lbrack {TP}_{01}^{1} \right\rbrack + \left\lbrack {TP}_{01}^{2} \right\rbrack}} & (23) \\{\left\lbrack P_{12} \right\rbrack_{0} = {\left\lbrack P_{12} \right\rbrack + \left\lbrack {TP}_{12}^{1} \right\rbrack + \left\lbrack {TP}_{12}^{2} \right\rbrack}} & (24) \\{\left\lbrack P_{02} \right\rbrack_{0} = {\left\lbrack P_{02} \right\rbrack + \left\lbrack {TP}_{02}^{1} \right\rbrack + \left\lbrack {TP}_{02}^{2} \right\rbrack}} & (25) \\\begin{matrix}{\lbrack T\rbrack_{0} = {\lbrack T\rbrack + \left\lbrack {TP}_{11}^{1} \right\rbrack + \left\lbrack {TP}_{01}^{1} \right\rbrack + \left\lbrack {TP}_{12}^{2} \right\rbrack + \left\lbrack {TP}_{02}^{2} \right\rbrack +}} \\{\left\lbrack {TP}_{11}^{2} \right\rbrack + \left\lbrack {TP}_{01}^{2} \right\rbrack + \left\lbrack {TP}_{12}^{1} \right\rbrack + \left\lbrack {TP}_{02}^{1} \right\rbrack} \\{= {\lbrack T\rbrack + \left( {\left\lbrack P_{11} \right\rbrack_{0} - \left\lbrack P_{11} \right\rbrack} \right) + \left( {\left\lbrack P_{01} \right\rbrack_{0} - \left\lbrack P_{01} \right\rbrack} \right) +}} \\{\left( {\left\lbrack P_{12} \right\rbrack_{0} - \left\lbrack P_{12} \right\rbrack} \right) + \left( {\left\lbrack P_{02} \right\rbrack_{0} - \left\lbrack P_{02} \right\rbrack} \right)}\end{matrix} & (26)\end{matrix}$Note that in these expressions [X]₀ denotes initial concentration of X,which is a free parameter, and [X] denotes its equilibriumconcentration. Consider the system consisting of equations (12)(19) andthe conservation rule equations (22)-(26). One has a system of 13polynomial equations (some quadratic, others linear) in 13 unknowns: X₁,. . . , X₉ (see (10)) and [P₁₁], [P₀₁], [P₁₂], [P₀₂], with 5 freeparameters: [P₁₁]₀, [P₀₁]₀, [P₁₂]₀, [P₀₂]₀, and [T]₀. Therefore, thisalgebraic system, when solved, yields the equilibrium concentrations.From these computed concentrations, we can evaluate the “matchto-mismatch ratio” for each probe:

$\left( \frac{\left\lbrack {TP}_{11}^{1} \right\rbrack + \left\lbrack {TP}_{11}^{2} \right\rbrack}{\left\lbrack {TP}_{01}^{1} \right\rbrack + \left\lbrack {TP}_{01}^{2} \right\rbrack} \right)_{{full}\mspace{14mu}{model}}\mspace{14mu}{and}\mspace{14mu}\left( \frac{\left\lbrack {TP}_{12}^{2} \right\rbrack + \left\lbrack {TP}_{12}^{1} \right\rbrack}{\left\lbrack {TP}_{02}^{2} \right\rbrack + \left\lbrack {TP}_{02}^{1} \right\rbrack} \right)_{{full}\mspace{14mu}{model}}$In order to examine the effects of competition between probes P₁₁ andP₁₂ on the signals for each of them, one now compares this situationwith the one where only P₁₁ and P₀₁ are present without P₁₂ or P₀₂, andvice versa. Hereinafter, the model introduced in this section as theFull Model and will compare its performance with the other two partialmodels, one consisting of P₁₁, P₀₁, and T only (referred to as Model I)and the other consisting of P₁₂, P₀₂, and T only (referred to as ModelII).Partial Model—Model I

This model consists of two probes P₁₁, P₀₁, and the target T only. Oneproceeds

as before by solving the algebraic system of equations to evaluate:

$\left( \frac{\left\lbrack {TP}_{11}^{1} \right\rbrack + \left\lbrack {TP}_{11}^{2} \right\rbrack}{\left\lbrack {TP}_{01}^{1} \right\rbrack + \left\lbrack {TP}_{01}^{2} \right\rbrack} \right)_{I}$Possible States:Consider the following states:(1) T(Target is unbound)(2) TP₁₁ ¹; (3) TP₀₁ ¹(Target is bound by “specific” hybridization.)(8) TP₁₁ ²; (9) TP₀₁ ²(Target is bound by “non-specific” hybridization)A state transition diagram for this partial model is depicted in FIG.3A. A set of reversible reactions operating between unbound and boundstates can be depicted as shown below.

$\begin{matrix}{{T + P_{11}}\underset{k_{2,1}}{\overset{k_{1,2}}{\rightleftarrows}}{TP}_{11}^{1}} & {{T + P_{11}}\underset{k_{8,1}}{\overset{k_{1,8}}{\rightleftarrows}}{TP}_{11}^{2}} \\{{T + P_{01}}\underset{k_{3,1}}{\overset{k_{1,3}}{\rightleftarrows}}{TP}_{01}^{1}} & {{T + P_{01}}\underset{k_{9,1}}{\overset{k_{1,9}}{\rightleftarrows}}{TP}_{01}^{2}}\end{matrix}$The following are the ordinary differential equations describing thedynamics of the system.

$\begin{matrix}\begin{matrix}{\frac{\mathbb{d}\lbrack T\rbrack}{\mathbb{d}t} = {{k_{2,1}\left\lbrack {TP}_{11}^{1} \right\rbrack} - {{k_{1,2}\lbrack T\rbrack}\left\lbrack P_{11} \right\rbrack} + {k_{3,1}\left\lbrack {TP}_{01}^{1} \right\rbrack} - {{k_{1,3}\lbrack T\rbrack}\left\lbrack P_{01} \right\rbrack} +}} \\{{k_{8,1}\left\lbrack {TP}_{11}^{2} \right\rbrack} - {{k_{1,8}\lbrack T\rbrack}\left\lbrack P_{11} \right\rbrack} + {k_{9,1}\left\lbrack {TP}_{01}^{2} \right\rbrack} - {{k_{1,9}\lbrack T\rbrack}\left\lbrack P_{01} \right\rbrack}}\end{matrix} & (27) \\{\frac{\mathbb{d}\left\lbrack {TP}_{11}^{1} \right\rbrack}{\mathbb{d}t} = {{{k_{1,2}\lbrack T\rbrack}\left\lbrack P_{11} \right\rbrack} - {k_{2,1}\left\lbrack {TP}_{11}^{1} \right\rbrack}}} & (28) \\{\frac{\mathbb{d}\left\lbrack {TP}_{01}^{1} \right\rbrack}{\mathbb{d}t} = {{{k_{1,3}\lbrack T\rbrack}\left\lbrack P_{01} \right\rbrack} - {k_{3,1}\left\lbrack {TP}_{01}^{1} \right\rbrack}}} & (29) \\{\frac{\mathbb{d}\left\lbrack {TP}_{11}^{2} \right\rbrack}{\mathbb{d}t} = {{{k_{1,8}\lbrack T\rbrack}\left\lbrack P_{11} \right\rbrack} - {k_{8,1}\left\lbrack {TP}_{11}^{2} \right\rbrack}}} & (30) \\{\frac{\mathbb{d}\left\lbrack {TP}_{01}^{2} \right\rbrack}{\mathbb{d}t} = {{{k_{1,9}\lbrack T\rbrack}\left\lbrack P_{01} \right\rbrack} - {k_{9,1}\left\lbrack {TP}_{01}^{2} \right\rbrack}}} & (31)\end{matrix}$Note that equations (28)-(31) are the same as equations (2), (3), (8),and (9) in the original full model system above, while equation (27)differs from (1),since it now involves only the states with probes P₁₁ and P₀₁.At equilibrium, d[.]/dt=0 for all substances, i.e., T, TP₁₁ ¹, TP₁₁ ²,TP₀₁ ¹, and TP₀₁ ², yielding:

$\begin{matrix}{K_{1}^{2} = \frac{\left\lbrack {TP}_{11}^{1} \right\rbrack}{\lbrack T\rbrack\left\lbrack P_{11} \right\rbrack}} & (32) \\{K_{1}^{3} = \frac{\left\lbrack {TP}_{01}^{1} \right\rbrack}{\lbrack T\rbrack\left\lbrack P_{01} \right\rbrack}} & (33) \\{K_{1}^{8} = \frac{\left\lbrack {TP}_{11}^{2} \right\rbrack}{\lbrack T\rbrack\left\lbrack P_{11} \right\rbrack}} & (34) \\{K_{1}^{9} = \frac{\left\lbrack {TP}_{01}^{1} \right\rbrack}{\lbrack T\rbrack\left\lbrack P_{01} \right\rbrack}} & (35)\end{matrix}$Since nothing else has changed in the thermodynamics, K₁ ^(j) computedfrom (21) are the same as before for jε{2, 3, 8, 9} and we have thefollowing conservation rules:

$\begin{matrix}{\left\lbrack P_{11} \right\rbrack_{0} = {\left\lbrack P_{11} \right\rbrack + \left\lbrack {TP}_{11}^{1} \right\rbrack + \left\lbrack {TP}_{11}^{2} \right\rbrack}} & (36) \\{\left\lbrack P_{01} \right\rbrack_{0} = {\left\lbrack P_{01} \right\rbrack + \left\lbrack {TP}_{01}^{1} \right\rbrack + \left\lbrack {TP}_{01}^{2} \right\rbrack}} & (37) \\\begin{matrix}{\lbrack T\rbrack_{0} = {\lbrack T\rbrack + \left\lbrack {TP}_{11}^{1} \right\rbrack + \left\lbrack {TP}_{01}^{1} \right\rbrack + \left\lbrack {TP}_{11}^{2} \right\rbrack + \left\lbrack {TP}_{01}^{2} \right\rbrack}} \\{= {\lbrack T\rbrack + \left( {\left\lbrack P_{11} \right\rbrack_{0} - \left\lbrack P_{11} \right\rbrack} \right) + \left( {\left\lbrack P_{01} \right\rbrack_{0} - \left\lbrack P_{01} \right\rbrack} \right)}}\end{matrix} & (38)\end{matrix}$In this case one has

-   -   7 variables (unknowns): [TP₁₁ ¹], [TP₁₁ ²], [TP₀₁ ¹], [TP₀₁ ²],        [P₁₁], [P₀₁], and [T]; and    -   7 polynomial equations: (32)-(35), (36), (37), and (38), with    -   3 free parameters [P₁₁]₀, [P₀₁]₀, and [T]₀.        Note that, for comparison with full model, the free parameters        will need to be scaled to retain the same initial        target-to-probe ratio.        Partial Model—Model II        This model consists of two probes P₁₂, P₀₂, and the target T        only. One proceeds as before by solving the algebraic system of        equations to evaluate:

$\left( \frac{\left\lbrack {TP}_{12}^{2} \right\rbrack + \left\lbrack {TP}_{12}^{1} \right\rbrack}{\left\lbrack {TP}_{02}^{2} \right\rbrack + \left\lbrack {TP}_{02}^{1} \right\rbrack} \right)_{11}$And now considers the following states:(1) T (Target is unbound.)(4) TP₁₂ ²; (5) TP₀₂ ²(Target is bound by “specific” hybridization)(6) TP₁₂ ¹; (7) TP₀₂ ¹(Target is bound by “non-specific” hybridization)A state transition diagram is shown in FIG. 3B. The set of reversiblereactions operating between unbound and bound states can be written asshown below.

$\begin{matrix}{{T + P_{12}}\underset{k_{4,1}}{\overset{k_{1,4}}{\rightleftarrows}}{TP}_{12}^{2}} & {{T + P_{12}}\underset{k_{8,1}}{\overset{k_{1,8}}{\rightleftarrows}}{TP}_{12}^{1}} \\{{T + P_{02}}\underset{k_{6,1}}{\overset{k_{1,6}}{\rightleftarrows}}{TP}_{02}^{2}} & {{T + P_{02}}\underset{k_{7,1}}{\overset{k_{1,7}}{\rightleftarrows}}{TP}_{02}^{1}}\end{matrix}$The following are the ordinary differential equations describing thedynamics of the system:

$\begin{matrix}\begin{matrix}{\frac{\mathbb{d}\lbrack T\rbrack}{\mathbb{d}t} = {{k_{4,1}\left\lbrack {TP}_{12}^{2} \right\rbrack} - {{k_{1,4}\lbrack T\rbrack}\left\lbrack P_{12} \right\rbrack} + {k_{5,1}\left\lbrack {TP}_{02}^{2} \right\rbrack} - {{k_{1,5}\lbrack T\rbrack}\left\lbrack P_{02} \right\rbrack} +}} \\{{k_{6,1}\left\lbrack {TP}_{12}^{1} \right\rbrack} - {{k_{1,6}\lbrack T\rbrack}\left\lbrack P_{12} \right\rbrack} + {k_{7,1}\left\lbrack {TP}_{02}^{1} \right\rbrack} - {{k_{1,7}\lbrack T\rbrack}\left\lbrack P_{02} \right\rbrack}}\end{matrix} & (39) \\{\frac{\mathbb{d}\left\lbrack {TP}_{12}^{2} \right\rbrack}{\mathbb{d}t} = {{{k_{1,4}\lbrack T\rbrack}\left\lbrack P_{12} \right\rbrack} - {k_{4,1}\left\lbrack {TP}_{02}^{2} \right\rbrack}}} & (40) \\{\frac{\mathbb{d}\left\lbrack {TP}_{02}^{2} \right\rbrack}{\mathbb{d}t} = {{{k_{1,5}\lbrack T\rbrack}\left\lbrack P_{02} \right\rbrack} - {k_{5,1}\left\lbrack {TP}_{02}^{2} \right\rbrack}}} & (41) \\{\frac{\mathbb{d}\left\lbrack {TP}_{12}^{1} \right\rbrack}{\mathbb{d}t} = {{{k_{1,6}\lbrack T\rbrack}\left\lbrack P_{12} \right\rbrack} - {k_{6,1}\left\lbrack {TP}_{12}^{1} \right\rbrack}}} & (42) \\{\frac{\mathbb{d}\left\lbrack {TP}_{02}^{1} \right\rbrack}{\mathbb{d}t} = {{{k_{1,7}\lbrack T\rbrack}\left\lbrack P_{02} \right\rbrack} - {k_{7,1}\left\lbrack {TP}_{02}^{1} \right\rbrack}}} & (43) \\\; & (44)\end{matrix}$Note that equations (40)-(43) are the same as equations (4), (5), (6),and (7) in the original system above, while equation (39) differs from(1), since it now involves only the states with probes P₁₂ and P₀₂. Atequilibrium, d[.]/dt=0 for all substances, i.e., T, TP₁₂ ², TP₁₂ ¹, TP₀₂², and TP₀₂ ¹, yielding:

$\begin{matrix}{K_{1}^{4} = \frac{\left\lbrack {TP}_{12}^{2} \right\rbrack}{\lbrack T\rbrack\left\lbrack P_{12} \right\rbrack}} & (45) \\{K_{1}^{5} = \frac{\left\lbrack {TP}_{02}^{2} \right\rbrack}{\lbrack T\rbrack\left\lbrack P_{02} \right\rbrack}} & (46) \\{K_{1}^{6} = \frac{\left\lbrack {TP}_{12}^{1} \right\rbrack}{\lbrack T\rbrack\left\lbrack P_{12} \right\rbrack}} & (47) \\{K_{1}^{7} = \frac{\left\lbrack {TP}_{02}^{1} \right\rbrack}{\lbrack T\rbrack\left\lbrack P_{02} \right\rbrack}} & (48)\end{matrix}$Again, since nothing else has changed in the thermodynamics, K₁ ^(j)computed from (21) are the same as before for jε{4, 5, 6, 7}, and wehave the following conservation rules:

$\begin{matrix}{\left\lbrack P_{12} \right\rbrack_{0} = {\left\lbrack P_{12} \right\rbrack + \left\lbrack {TP}_{12}^{2} \right\rbrack + \left\lbrack {TP}_{12}^{1} \right\rbrack}} & (49) \\{\left\lbrack P_{02} \right\rbrack_{0} = {\left\lbrack P_{02} \right\rbrack + \left\lbrack {TP}_{02}^{2} \right\rbrack + \left\lbrack {TP}_{02}^{1} \right\rbrack}} & (50) \\\begin{matrix}{\lbrack T\rbrack_{0} = {\lbrack T\rbrack + \left\lbrack {TP}_{12}^{2} \right\rbrack + \left\lbrack {TP}_{02}^{2} \right\rbrack + \left\lbrack {TP}_{12}^{1} \right\rbrack + \left\lbrack {TP}_{02}^{1} \right\rbrack}} \\{= {\lbrack T\rbrack + \left( {\left\lbrack P_{12} \right\rbrack_{0} - \left\lbrack P_{12} \right\rbrack} \right) + \left( {\left\lbrack P_{02} \right\rbrack_{0} - \left\lbrack P_{02} \right\rbrack} \right)}}\end{matrix} & (51)\end{matrix}$In this case, one also has:7 variables: [TP₁₂ ²], [TP₁₂ ¹], [TP₀₂ ²], [TP₀₂ ¹], [P₁₂], [P₀₂], and[T]; and7 equations: (45)-(48), (49), (50), and (51), with3 free parameters [P₁₂]₀, [P₀₂]₀, and [T]₀.As above, the parameters will need to be scaled. In practice, once theexact nucleotide sequences of T, P₁₁, P₀₁, P₁₂, and P₀₂ are determinedfrom the needs of the biological assay, one can compute K₁ ^(j)explicitly, and then solve for the unknowns in all three setups: FullModel, Model I, and Model II. With these computed ratio values, one isready to evaluate and compare the models in order to discern the effectsof competition:

$\left( \frac{P_{11}}{P_{01}} \right)_{full}\mspace{14mu}{{vs}.\mspace{14mu}\left( \frac{P_{11}}{P_{01}} \right)_{I}}\mspace{14mu}{{and}\left( \frac{P_{12}}{P_{02}} \right)}_{full}\mspace{14mu}{{vs}.\mspace{14mu}\left( \frac{P_{12}}{P_{02}} \right)_{II}}$Full ModelIn order to simplify the algebraic system of equation, one can renamethe unknown variables as follows:

$\begin{matrix}{X_{1} = \lbrack T\rbrack} & \; & \; \\{X_{2} = \left\lbrack {TP}_{11}^{1} \right\rbrack} & {X_{6} = \left\lbrack {TP}_{12}^{1} \right\rbrack} & {Y_{1} = \left\lbrack P_{11} \right\rbrack} \\{X_{3} = \left\lbrack {TP}_{01}^{1} \right\rbrack} & {X_{7} = \left\lbrack {TP}_{02}^{1} \right\rbrack} & {Y_{2} = \left\lbrack P_{01} \right\rbrack} \\{X_{4} = \left\lbrack {TP}_{12}^{2} \right\rbrack} & {X_{8} = \left\lbrack {TP}_{11}^{2} \right\rbrack} & {Y_{3} = \left\lbrack P_{12} \right\rbrack} \\{X_{5} = \left\lbrack {TP}_{02}^{2} \right\rbrack} & {X_{9} = \left\lbrack {TP}_{01}^{2} \right\rbrack} & {Y_{4} = \left\lbrack P_{02} \right\rbrack}\end{matrix}$The constant parameters in the system are initially left in theirsymbolic form.

K₁²; K₁³; K₁⁴; K₁⁵; K₁⁶; K₁⁷; K₁⁸; K₁⁹; $\begin{matrix}{{a_{0} = \left\lbrack P_{11} \right\rbrack_{0}};} & {{b_{0} = \left\lbrack P_{01} \right\rbrack_{0}};} \\{{c_{0} = \left\lbrack P_{12} \right\rbrack_{0}};} & {{d_{0} = \left\lbrack P_{02} \right\rbrack_{0}};} \\{e_{0} = {\lbrack T\rbrack_{0}.}} & \;\end{matrix}$Equations (12)-(19) and (22)-(26) can now be rewritten in terms of{X_(i); Y_(j)} as follows.

$\begin{matrix}\left. \begin{matrix}{\left\lbrack {TP}_{11}^{1} \right\rbrack = \left. {{K_{1}^{2}\lbrack T\rbrack}\left\lbrack P_{11} \right\rbrack}\Longrightarrow \right.} & {X_{2} = {K_{1}^{2}X_{1}Y_{1}}} \\{\left\lbrack {TP}_{01}^{1} \right\rbrack = \left. {{K_{1}^{3}\lbrack T\rbrack}\left\lbrack P_{01} \right\rbrack}\Longrightarrow \right.} & {X_{3} = {K_{1}^{3}X_{1}Y_{2}}} \\{\left\lbrack {TP}_{12}^{2} \right\rbrack = \left. {{K_{1}^{4}\lbrack T\rbrack}\left\lbrack P_{12} \right\rbrack}\Longrightarrow \right.} & {X_{4} = {K_{1}^{4}X_{1}Y_{3}}} \\{\left\lbrack {TP}_{02}^{2} \right\rbrack = \left. {{K_{1}^{5}\lbrack T\rbrack}\left\lbrack P_{02} \right\rbrack}\Longrightarrow \right.} & {X_{5} = {K_{1}^{5}X_{1}Y_{4}}} \\{\left\lbrack {TP}_{12}^{1} \right\rbrack = \left. {{K_{1}^{6}\lbrack T\rbrack}\left\lbrack P_{12} \right\rbrack}\Longrightarrow \right.} & {X_{6} = {K_{1}^{6}X_{1}Y_{3}}} \\{\left\lbrack {TP}_{02}^{1} \right\rbrack = \left. {{K_{1}^{7}\lbrack T\rbrack}\left\lbrack P_{02} \right\rbrack}\Longrightarrow \right.} & {X_{7} = {K_{1}^{7}X_{1}Y_{4}}} \\{\left\lbrack {TP}_{11}^{2} \right\rbrack = \left. {{K_{1}^{8}\lbrack T\rbrack}\left\lbrack P_{11} \right\rbrack}\Longrightarrow \right.} & {X_{8} = {K_{1}^{8}X_{1}Y_{1}}} \\{\left\lbrack {TP}_{01}^{2} \right\rbrack = \left. {{K_{1}^{9}\lbrack T\rbrack}\left\lbrack P_{01} \right\rbrack}\Longrightarrow \right.} & {X_{9} = {K_{1}^{9}X_{1}Y_{2}}} \\{\left\lbrack P_{11} \right\rbrack_{0} = {\left\lbrack P_{11} \right\rbrack + \left\lbrack {TP}_{11}^{1} \right\rbrack + \left. \left\lbrack {TP}_{11}^{2} \right\rbrack\Longrightarrow \right.}} & {a_{0} = {X_{2} + X_{8} + Y_{1}}} \\{\left\lbrack P_{01} \right\rbrack_{0} = {\left\lbrack P_{01} \right\rbrack + \left\lbrack {TP}_{01}^{1} \right\rbrack + \left. \left\lbrack {TP}_{01}^{2} \right\rbrack\Longrightarrow \right.}} & {b_{0} = {X_{3} + X_{9} + Y_{2}}} \\{\left\lbrack P_{12} \right\rbrack_{0} = {\left\lbrack P_{12} \right\rbrack + \left\lbrack {TP}_{12}^{1} \right\rbrack + \left. \left\lbrack {TP}_{12}^{2} \right\rbrack\Longrightarrow \right.}} & {c_{0} = {X_{4} + X_{6} + Y_{3}}} \\{\left\lbrack P_{02} \right\rbrack_{0} = {\left\lbrack P_{02} \right\rbrack + \left\lbrack {TP}_{02}^{1} \right\rbrack + \left. \left\lbrack {TP}_{02}^{2} \right\rbrack\Longrightarrow \right.}} & {d_{0} = {X_{5} + X_{7} + Y_{4}}} \\{\lbrack T\rbrack_{0} = {\lbrack T\rbrack + \left\lbrack {TP}_{11}^{1} \right\rbrack + \left\lbrack {TP}_{01}^{1} \right\rbrack +}} & {e_{0} = {X_{1} + X_{2} + X_{3} +}} \\{\left\lbrack {TP}_{12}^{2} \right\rbrack + \left\lbrack {TP}_{02}^{2} \right\rbrack + \left\lbrack {TP}_{11}^{2} \right\rbrack +} & {X_{4} + X_{5} + X_{6} +} \\{\left\lbrack {TP}_{01}^{2} \right\rbrack + \left\lbrack {TP}_{12}^{1} \right\rbrack + \left. \left\lbrack {TP}_{02}^{1} \right\rbrack\Longrightarrow \right.} & {X_{7} + X_{8} + X_{9}}\end{matrix} \right\} & (52)\end{matrix}$Model INow, we consider a system of algebraic equations representing theconcentrations at equilibrium and involving unknown variables X₁, X₂,X₃, X₈, X₉, Y₁, and Y₂, and constant parameters K₁ ², K₁ ³, K₁ ⁸, K₁ ⁹,a₀, b₀, and e₀. Thus, in a manner analogous to that derived for the fullmodel in the previous section, one may rewrite the equations (32), (33),(34), (35), (36), (37), and (38) in terms of {X_(i); Y_(j)}, as shownbelow.

$\begin{matrix}\left. \begin{matrix}{X_{2} = {K_{1}^{2}X_{1}Y_{1}}} \\{X_{3} = {K_{1}^{3}X_{1}Y_{2}}} \\{X_{8} = {K_{1}^{8}X_{1}Y_{1}}} \\{X_{9} = {K_{1}^{9}X_{1}Y_{2}}} \\{a_{0} = {\left\lbrack P_{11} \right\rbrack_{0} = {X_{2} + X_{8} + Y_{1}}}} \\{b_{0} = {\left\lbrack P_{01} \right\rbrack_{0} = {X_{3} + X_{9} + Y_{2}}}} \\{e_{0} = {\lbrack T\rbrack_{0} = {X_{1} + X_{2} + X_{3} + X_{8} + X_{9}}}}\end{matrix} \right\} & (53)\end{matrix}$Model IINext, consider a system of algebraic equations representing theconcentrations at equilibrium and involving unknown variables X₁, X₄,X₅, X₆, X₇, Y₃, and Y₄, and constant parameters K₁ ⁴, K₁ ⁵, K₁ ⁶, K₁ ⁷,c₀, d₀, and e₀. Once again one may rewrite the equations (45), (46),(47), (48), (49), (50), and (51) in terms of {X_(i); Y_(j)}, as shownbelow.

$\begin{matrix}\left. \begin{matrix}{X_{4} = {K_{1}^{4}X_{1}Y_{3}}} \\{X_{5} = {K_{1}^{5}X_{1}Y_{4}}} \\{X_{6} = {K_{1}^{6}X_{1}Y_{3}}} \\{X_{7} = {K_{1}^{7}X_{1}Y_{4}}} \\{c_{0} = {\left\lbrack P_{12} \right\rbrack_{0} = {X_{4} + X_{6} + Y_{3}}}} \\{d_{0} = {\left\lbrack P_{02} \right\rbrack_{0} = {X_{5} + X_{7} + Y_{4}}}} \\{e_{0} = {\lbrack T\rbrack_{0} = {X_{1} + X_{4} + X_{5} + X_{6} + X_{7}}}}\end{matrix} \right\} & (54)\end{matrix}$Note that with the exception of the conservation rules for [T] (i.e.,the last equations in (52), (53), and (54)) under the different models,one has (52)=(53)∪(54).System Reduction—Model IStarting with (53), one may obtain the following linear equalities:Y ₁ =a ₀ −X ₂ −X ₈  (55)Y ₂ =b ₀ −X ₃ −X ₉  (56)Furthermore, since

$\begin{matrix}{\frac{X_{2}}{X_{8}} = {\frac{K_{1}^{2}X_{1}Y_{1}}{K_{1}^{8}X_{1}Y_{1}} = {\left. \frac{K_{1}^{2}}{K_{1}^{8}}\Longrightarrow X_{8} \right. = {\frac{K_{1}^{8}}{K_{1}^{2}}X_{2}}}}} & (57) \\{\frac{X_{3}}{X_{9}} = {\frac{K_{1}^{3}X_{1}Y_{2}}{K_{1}^{9}X_{1}Y_{2}} = {\left. \frac{K_{1}^{3}}{K_{1}^{9}}\Longrightarrow X_{9} \right. = {\frac{K_{1}^{9}}{K_{1}^{3}}X_{3}}}}} & (58)\end{matrix}$one may simplify to obtain:

$\begin{matrix}{\begin{matrix}{X_{2} = {{K_{1}^{2}X_{1}Y_{1}} = {K_{1}^{2}{X_{1}\left( {a_{0} - X_{2} - X_{8}} \right)}}}} \\{= {K_{1}^{2}{X_{1}\left( {a_{0} - X_{2} - {\frac{K_{1}^{8}}{K_{1}^{2}}X_{2}}} \right)}}} \\{= {K_{1}^{2}{X_{1}\left( {a_{0} - {X_{2}\left\lbrack {1 + \frac{K_{1}^{8}}{K_{1}^{2}}} \right\rbrack}} \right)}}} \\{= {{K_{1}^{2}X_{1}a_{0}} - {K_{1}^{2}X_{1}{X_{2}\left\lbrack {1 + \frac{K_{1}^{8}}{K_{1}^{2}}} \right\rbrack}}}} \\{= {{K_{1}^{2}X_{1}a_{0}} - {X_{1}{X_{2}\left\lbrack {K_{1}^{2} + K_{1}^{8}} \right\rbrack}}}} \\{{X_{2} + {X_{1}{X_{2}\left\lbrack {K_{1}^{2} + K_{1}^{8}} \right\rbrack}}} = {a_{0}K_{1}^{2}X_{1}}} \\{\therefore{X_{2} = \frac{a_{0}K_{1}^{2}X_{1}}{1 + {X_{1}\left( {K_{1}^{2} + K_{1}^{8}} \right)}}}}\end{matrix}{and}} & (59) \\\begin{matrix}{X_{3} = {{K_{1}^{9}X_{1}Y_{2}} = {K_{1}^{3}{X_{1}\left( {b_{0} - X_{3} - X_{9}} \right)}}}} \\{= {K_{1}^{9}{X_{1}\left( {b_{0} - X_{3} - {\frac{K_{1}^{9}}{K_{1}^{3}}X_{3}}} \right)}}} \\{= {K_{1}^{9}{X_{1}\left( {b_{0} - {X_{3}\left\lbrack {1 + \frac{K_{1}^{9}}{K_{1}^{3}}} \right\rbrack}} \right)}}} \\{= {{K_{1}^{9}X_{1}b_{0}} - {K_{1}^{3}X_{1}{X_{3}\left\lbrack {1 + \frac{K_{1}^{9}}{K_{1}^{3}}} \right\rbrack}}}} \\{= {{K_{1}^{9}X_{1}b_{0}} - {X_{1}{X_{3}\left\lbrack {K_{1}^{3} + K_{1}^{9}} \right\rbrack}}}} \\{{X_{3} + {X_{1}{X_{3}\left\lbrack {K_{1}^{3} + K_{1}^{9}} \right\rbrack}}} = {b_{0}K_{1}^{3}X_{1}}} \\{\therefore{X_{3} = \frac{b_{0}K_{1}^{3}X_{1}}{1 + {X_{1}\left( {K_{1}^{3} + K_{1}^{9}} \right)}}}}\end{matrix} & (60)\end{matrix}$We also obtain, from (57),

$\begin{matrix}\begin{matrix}{X_{8} = {{\frac{K_{1}^{8}}{K_{1}^{2}}X_{2}} = {\frac{K_{1}^{8}}{K_{1}^{2}}\frac{a_{0}K_{1}^{2}X_{1}}{1 + {X_{1}\left( {K_{1}^{2} + K_{1}^{8}} \right)}}}}} \\{= {\frac{a_{0}K_{1}^{8}X_{1}}{1 + {X_{1}\left( {K_{1}^{2} + K_{1}^{8}} \right)}} = X_{8}}}\end{matrix} & (61)\end{matrix}$and from (58):

$\begin{matrix}\begin{matrix}{X_{9} = {{\frac{K_{1}^{9}}{K_{1}^{3}}X_{3}} = {\frac{K_{1}^{9}}{K_{1}^{3}}\frac{b_{0}K_{1}^{3}X_{1}}{1 + {X_{1}\left( {K_{1}^{3} + K_{1}^{9}} \right)}}}}} \\{= {\frac{b_{0}K_{1}^{9}X_{1}}{1 + {X_{1}\left( {K_{1}^{3} + K_{1}^{9}} \right)}} = X_{9}}}\end{matrix} & (62)\end{matrix}$Finally, equations (59), (60), (61), and (62) can be solved to expressX₂, X₃, X₈, and X₉, respectively, in terms of X₁. Now, from (55), (57),and (59), one derives:

$\begin{matrix}\begin{matrix}{Y_{1} = {{a_{0} - X_{2} - X_{8}} = {a_{0} - {X_{2}\left( {1 + \frac{K_{1}^{8}}{K_{1}^{2}}} \right)}}}} \\{= {a_{0} - {\frac{a_{0}K_{1}^{2}X_{1}}{1 + {X_{1}\left( {K_{1}^{2} + K_{1}^{8}} \right)}}\left( {1 + \frac{K_{1}^{8}}{K_{1}^{2}}} \right)}}} \\{= {a_{0} - \frac{a_{0}{X_{1}\left( {K_{1}^{2} + K_{1}^{8}} \right)}}{1 + {X_{1}\left( {K_{1}^{2} + K_{1}^{8}} \right)}}}} \\{= {a_{0}\left\lbrack \frac{1 + {X_{1}\left( {K_{1}^{2} + K_{1}^{8}} \right)} - {X_{1}\left( {K_{1}^{2} + K_{1}^{8}} \right)}}{1 + {X_{1}\left( {K_{1}^{2} + K_{1}^{8}} \right)}} \right\rbrack}} \\{= \frac{a_{0}}{1 + {X_{1}\left( {K_{1}^{2} + K_{1}^{8}} \right)}}} \\{\therefore{Y_{1} = \frac{a_{0}}{1 + {X_{1}\left( {K_{1}^{2} + K_{1}^{8}} \right)}}}}\end{matrix} & (63)\end{matrix}$Similarly, one derives:

$\begin{matrix}\begin{matrix}{Y_{2} = {{b_{0} - X_{3} - X_{9}} = {b_{0} - {X_{3}\left( {1 + \frac{K_{1}^{9}}{K_{1}^{3}}} \right)}}}} \\{= {b_{0} - {\frac{b_{0}K_{1}^{3}X_{1}}{1 + {X_{1}\left( {K_{1}^{3} + K_{1}^{9}} \right)}}\left( {1 + \frac{K_{1}^{9}}{K_{1}^{3}}} \right)}}} \\{= {b_{0} - \frac{b_{0}{X_{1}\left( {K_{1}^{3} + K_{1}^{9}} \right)}}{1 + {X_{1}\left( {K_{1}^{3} + K_{1}^{9}} \right)}}}} \\{= {b_{0}\left\lbrack \frac{1 + {X_{1}\left( {K_{1}^{3} + K_{1}^{9}} \right)} - {X_{1}\left( {K_{1}^{3} + K_{1}^{9}} \right)}}{1 + {X_{1}\left( {K_{1}^{3} + K_{1}^{9}} \right)}} \right\rbrack}} \\{= \frac{b_{0}}{1 + {X_{1}\left( {K_{1}^{3} + K_{1}^{9}} \right)}}} \\{\therefore{Y_{2} = \frac{b_{0}}{1 + {X_{1}\left( {K_{1}^{3} + K_{1}^{9}} \right)}}}}\end{matrix} & (64)\end{matrix}$A final simplification yields a univariate rational function only in X₁equating to a constant e₀:

$\begin{matrix}\begin{matrix}{e_{0} = {X_{1} + X_{2} + X_{3} + X_{8} + {X_{9}\mspace{14mu}\left( {{by}\mspace{14mu}(53)} \right)}}} \\{= {X_{1} + {X_{1}\frac{a_{0}K_{1}^{2}}{1 + {X_{1}\left( {K_{1}^{2} + K_{1}^{8}} \right)}}} +}} \\{{X_{1}\frac{b_{0}K_{1}^{3}}{1 + {X_{1}\left( {K_{1}^{3} + K_{1}^{9}} \right)}}\mspace{14mu}\left( {{{by}\mspace{14mu}(59)};(60)} \right)} +} \\{{X_{1}\frac{a_{0}K_{1}^{8}}{1 + {X_{1}\left( {K_{1}^{2} + K_{1}^{8}} \right)}}} +} \\{X_{1}\frac{b_{0}K_{1}^{9}}{1 + {X_{1}\left( {K_{1}^{3} + K_{1}^{9}} \right)}}\mspace{14mu}\left( {{{by}\mspace{14mu}(61)};(62)} \right)} \\{e_{0} = {X_{1}\left\lbrack {1 + {a_{0}\frac{K_{1}^{2} + K_{1}^{8}}{1 + {X_{1}\left( {K_{1}^{2} + K_{1}^{8}} \right)}}} + {b_{0}\frac{K_{1}^{3} + K_{1}^{9}}{1 + {X_{1}\left( {K_{1}^{3} + K_{1}^{9}} \right)}}}} \right\rbrack}}\end{matrix} & (65)\end{matrix}$Since the terms (K₁ ²+K₁ ⁸) and (K₁ ³+K₁ ⁹) appear frequently, in orderto express the preceding equations in a simpler form, introduced is ashort-hand notation shown below. Let

s₂₈ ≡ K₁² + K₁⁸; s₃₉ ≡ K₁³ + K₁⁹; and  x ≡ X₁.In simplified form, the equation (65) becomes:

$\begin{matrix}{{{\left. {{{x\left( {1 + {a_{0}\frac{s_{28}}{1 + {s_{28}x}}} + {b_{0}\frac{s_{39}}{1 + {s_{39}x}}}} \right)} = e_{0}}{{x\left( \frac{\begin{matrix}\left( {{\left( {1 + {s_{28}x}} \right)\left( {1 + {s_{39}x}} \right)} + {a_{0}{s_{28}\left( {1 + {s_{39}x}} \right)}} +} \right. \\\left. {b_{0}{s_{39}\left( {1 + {s_{28}x}} \right)}} \right)\end{matrix}}{\left( {\left( {1 + {s_{28}x}} \right)\left( {1 + {s_{39}x}} \right)} \right)} \right)} = e_{0}}{{{x\left( {{\left( {1 + {s_{28}x}} \right)\left( {1 + {s_{39}x}} \right)} + {a_{0}{s_{28}\left( {1 + {s_{39}x}} \right)}} + {b_{0}{s_{39}\left( {1 + {s_{28}x}} \right)}}} \right)} = {{e_{0}\left( {1 + {s_{28}x}} \right)}\left( {1 + {s_{39}x}} \right)}};}{{\left( {s_{28}s_{39}} \right)x^{3}} + {\left( {s_{28} + s_{39} + {s_{28}{s_{39}\left\lbrack {a_{0} + b_{0} - e_{0}} \right\rbrack}}} \right)x^{2}} + 1 + {s_{28}\left\lbrack {a_{0} - e_{0}} \right\rbrack} + {s_{39}\left\lbrack {b_{0} - e_{0}} \right\rbrack}}} \right)x} - e_{0}} = 0} & (66)\end{matrix}$

Now the cubic polynomial equation (66) must be solved for the unknownχ=X₁, and then the solution can be substituted into (59)-(64) in orderto solve for the rest of the variables. One may obtain the solutions intheir symbolic form using MATHEMATICA computer algebra system (seeWolfram, S. The Mathematica Book. Cambridge University Press, 4^(th)edition, (1999)), as the three possible roots may be easily expressed inradicals. More to the point, one only needs to solve for

$\begin{matrix}{\begin{matrix}{\left( \frac{P_{11}}{P_{01}} \right)_{I} = \left( \frac{\left\lbrack {TP}_{11}^{1} \right\rbrack + \left\lbrack {TP}_{11}^{2} \right\rbrack}{\left\lbrack {TP}_{01}^{1} \right\rbrack + \left\lbrack {TP}_{01}^{2} \right\rbrack} \right)_{1}} \\{= \frac{X_{2} + X_{8}}{X_{3} + X_{9}}} \\{= {\left( {\frac{a_{0}K_{1}^{2}x}{1 + {s_{28}x}} + \frac{a_{0}K_{1}^{8}x}{1 + {s_{28}x}}} \right)/\left( {\frac{b_{0}K_{1}^{3}x}{1 + {s_{39}x}} + \frac{b_{0}K_{1}^{9}x}{1 + {s_{39}x}}} \right)}}\end{matrix}{or}\begin{matrix}{\left( \frac{P_{11}}{P_{01}} \right)_{I} = {\left( \frac{a_{0}s_{28}x}{1 + {s_{28}x}} \right)/\left( \frac{b_{0}s_{39}x}{1 + {s_{39}x}} \right)}} \\{= {\frac{a_{0}}{b_{0}}\frac{s_{28}}{s_{39}}\frac{1 + {s_{39}x}}{1 + {s_{28}x}}}} \\{= {\frac{a_{0}}{b_{0}}\frac{s_{28}}{s_{39}}\frac{s_{39} + {1/x}}{s_{28} + {1/x}}}}\end{matrix}} & (67)\end{matrix}$where χ is at solution of (66).System Reduction—Model IIAs before, starting with (54), one obtains the following linearequalities:Y ₃ =c ₀ −X ₄ −X ₆  (68)Y ₄ =d ₀ −X ₅ −X ₇  (69)Since

$\begin{matrix}{\frac{X_{4}}{X_{6}} = {\frac{K_{1}^{4}X_{1}Y_{3}}{K_{1}^{6}X_{1}Y_{3}} = {\left. \frac{K_{1}^{4}}{K_{1}^{6}}\Longrightarrow X_{6} \right. = {\frac{K_{1}^{6}}{K_{1}^{4}}X_{4}}}}} & (70) \\{{\frac{X_{5}}{X_{7}} = {\frac{K_{1}^{6}X_{1}Y_{4}}{K_{1}^{7}X_{1}Y_{4}} = {\left. \frac{K_{1}^{5}}{K_{1}^{7}}\Longrightarrow X_{7} \right. = {\frac{K_{1}^{7}}{K_{1}^{5}}X_{5}}}}}{{we}\mspace{14mu}{obtain}}} & (71) \\{\begin{matrix}{X_{4} = {{K_{1}^{4}X_{1}Y_{3}} = {K_{1}^{4}{X_{1}\left( {c_{0} - {X_{4}\left\lbrack {1 + \frac{K_{1}^{6}}{K_{1}^{4}}} \right\rbrack}} \right)}}}} \\{= {{K_{1}^{4}X_{1}c_{0}} - {X_{1}{X_{4}\left( {K_{1}^{4} + K_{1}^{6}} \right)}}}} \\{\therefore{X_{4} = \frac{c_{0}K_{1}^{4}X_{1}}{1 + {X_{1}\left( {K_{1}^{4} + K_{1}^{6}} \right)}}}}\end{matrix}{and}} & (72) \\\begin{matrix}{X_{5} = {K_{1}^{5}X_{1}Y_{4}}} \\{= {K_{1}^{5}{X_{1}\left( {d_{0} - {X_{5}\left\lbrack {1 + \frac{K_{1}^{7}}{K_{1}^{5}}} \right\rbrack}} \right)}}} \\{= {{K_{1}^{5}X_{1}d_{0}} - {X_{1}{X_{5}\left( {K_{1}^{5} + K_{1}^{7}} \right)}}}} \\{\therefore{X_{5} = \frac{d_{0}K_{1}^{5}X_{1}}{1 + {X_{1}\left( {K_{1}^{5} + K_{1}^{7}} \right)}}}}\end{matrix} & (73)\end{matrix}$Furthermore, from (70) and (72), we obtain

$\begin{matrix}\begin{matrix}{X_{6} = {\frac{K_{1}^{6}}{K_{1}^{4}}X_{4}}} \\{= {\frac{K_{1}^{6}}{K_{1}^{4}}\frac{c_{0}K_{1}^{4}X_{1}}{1 + {X_{1}\left( {K_{1}^{4} + K_{1}^{6}} \right)}}}} \\{= {\frac{c_{0}K_{1}^{6}X_{1}}{1 + {X_{1}\left( {K_{1}^{4} + K_{1}^{6}} \right)}} = X_{6}}}\end{matrix} & (74)\end{matrix}$and from (71) and (73),

$\begin{matrix}\begin{matrix}{X_{7} = {\frac{K_{1}^{7}}{K_{1}^{5}}X_{5}}} \\{= {\frac{K_{1}^{7}}{K_{1}^{5}}\frac{d_{0}K_{1}^{6}X_{1}}{1 + {X_{1}\left( {K_{1}^{6} + K_{1}^{7}} \right)}}}} \\{= {\frac{d_{0}K_{1}^{7}X_{1}}{1 + {X_{1}\left( {K_{1}^{6} + K_{1}^{7}} \right)}} = X_{7}}}\end{matrix} & (75)\end{matrix}$Finally, equations (72) (73), (74), and (75) can be solved to expressX₄, X₅, X₆, and X₇, respectively, in terms of X₁.

From (68), (70), and (72), we derive

$\begin{matrix}\begin{matrix}{Y_{3} = {c_{0} - X_{4} - X_{6}}} \\{= {c_{0} - {X_{4}\left( {1 + \frac{K_{1}^{6}}{K_{1}^{4}}} \right)}}} \\{= {c_{0} - {\frac{c_{0}K_{1}^{4}X_{1}}{1 + {X_{1}\left( {K_{1}^{4} + K_{1}^{6}} \right)}}\left( {1 + \frac{K_{1}^{6}}{K_{1}^{4}}} \right)}}} \\{= {c_{0} - \frac{c_{0}{X_{1}\left( {K_{1}^{4} + K_{1}^{6}} \right)}}{1 + {X_{1}\left( {K_{1}^{4} + K_{1}^{6}} \right)}}}} \\{= {c_{0}\left\lbrack \frac{1 + {X_{1}\left( {K_{1}^{4} + K_{1}^{6}} \right)} - {X_{1}\left( {K_{1}^{4} + K_{1}^{6}} \right)}}{1 + {X_{1}\left( {K_{1}^{4} + K_{1}^{6}} \right)}} \right\rbrack}} \\{= \frac{c_{0}}{1 + {X_{1}\left( {K_{1}^{4} + K_{1}^{6}} \right)}}} \\{\therefore{Y_{3} = \frac{c_{0}}{1 + {X_{1}\left( {K_{1}^{4} + K_{1}^{6}} \right)}}}}\end{matrix} & (76)\end{matrix}$Similarly, we derive

$\begin{matrix}\begin{matrix}{Y_{4} = {d_{0} - X_{5} - X_{7}}} \\{= {d_{0} - {X_{5}\left( {1 + \frac{K_{1}^{7}}{K_{1}^{5}}} \right)}}} \\{= {d_{0} - {\frac{d_{0}K_{1}^{5}X_{1}}{1 + {X_{1}\left( {K_{1}^{5} + K_{1}^{7}} \right)}}\left( {1 + \frac{K_{1}^{7}}{K_{1}^{5}}} \right)}}} \\{= {d_{0} - \frac{d_{0}{X_{1}\left( {K_{1}^{5} + K_{1}^{7}} \right)}}{1 + {X_{1}\left( {K_{1}^{5} + K_{1}^{7}} \right)}}}} \\{= {d_{0}\left\lbrack \frac{1 + {X_{1}\left( {K_{1}^{5} + K_{1}^{7}} \right)} - {X_{1}\left( {K_{1}^{5} + K_{1}^{7}} \right)}}{1 + {X_{1}\left( {K_{1}^{5} + K_{1}^{7}} \right)}} \right\rbrack}} \\{= \frac{d_{0}}{1 + {X_{1}\left( {K_{1}^{5} + K_{1}^{7}} \right)}}} \\{\therefore{Y_{4} = \frac{d_{0}}{1 + {X_{1}\left( {K_{1}^{5} + K_{1}^{7}} \right)}}}}\end{matrix} & (77)\end{matrix}$Putting it all together, we derive the univariate rational equation forX₁.or

$\begin{matrix}\begin{matrix}{e_{0} = {X_{1} + X_{4} + X_{5} + X_{6} + {X_{7}\mspace{14mu}\left( {{by}\mspace{14mu}(54)} \right)}}} \\{= {X_{1} + {X_{1}\frac{c_{0}K_{1}^{4}}{1 + {X_{1}\left( {K_{1}^{4} + K_{1}^{6}} \right)}}} +}} \\{{X_{1}\frac{d_{0}K_{1}^{5}}{1 + {X_{1}\left( {K_{1}^{5} + K_{1}^{7}} \right)}}\mspace{14mu}\left( {{{by}\mspace{14mu}(72)},(73)} \right)} +} \\{{X_{1}\frac{c_{0}K_{1}^{6}}{1 + {X_{1}\left( {K_{1}^{4} + K_{1}^{6}} \right)}}} +} \\{X_{1}\frac{d_{0}K_{1}^{7}}{1 + {X_{1}\left( {K_{1}^{5} + K_{1}^{7}} \right)}}\mspace{14mu}\left( {{{by}\mspace{14mu}(74)},(75)} \right)} \\{e_{0} = {X_{1}\left\lbrack {1 + {c_{0}\frac{K_{1}^{4} + K_{1}^{6}}{1 + {X_{1}\left( {K_{1}^{4} + K_{1}^{6}} \right)}}} + {d_{0}\frac{K_{1}^{5} + K_{1}^{7}}{1 + {X_{1}\left( {K_{1}^{5} + K_{1}^{7}} \right)}}}} \right\rbrack}}\end{matrix} & (78)\end{matrix}$As before, we abbreviate the terms (K₁ ⁴+K₁ ⁶) and (K₁ ⁵+K₁ ⁷) byshort-hand notation, shown below. Let

s₄₆ ≡ K₁⁴ + K₁⁶; s₅₇ ≡ K₁⁵ + K₁⁷; and  y ≡ X₁.Note that, in order to avoid confusion a different abbreviation forX_(i) (i.e., y), was introduced intentionally since the equation to besolved in this case differs from equation (66). Then (78) can beexpressed as:

$\begin{matrix}{{{y\left( {1 + {c_{0}\frac{s_{46}}{1 + {s_{46}y}}} + {d_{0}\frac{s_{57}}{1 + {s_{57}y}}}} \right)} = e_{0}}{{y\left( \frac{{\left( {1 + {s_{46}y}} \right)\left( {1 + {s_{57}y}} \right)} + {c_{0}{s_{46}\left( {1 + {s_{57}y}} \right)}} + {d_{0}{s_{57}\left( {1 + {s_{46}y}} \right)}}}{\left( {1 + {s_{46}y}} \right)\left( {1 + {s_{57}y}} \right)} \right)} = e_{0}}{{{y\left( {{\left( {1 + {s_{46}y}} \right)\left( {1 + {s_{57}y}} \right)} + {c_{0}{s_{46}\left( {1 + {s_{57}y}} \right)}} + {d_{0}{s_{57}\left( {1 + {s_{46}y}} \right)}}} \right)} = {{e_{0}\left( {1 + {s_{46}y}} \right)}\left( {1 + {s_{57}y}} \right)}};}{{{{{or}\mspace{14mu}\left( {s_{46}s_{57}} \right)}y^{3}} + {\left( {s_{46} + s_{57} + {s_{46}{s_{57}\left\lbrack {c_{0} + d_{0} - e_{0}} \right\rbrack}}} \right)y^{2}} + {\left( {1 + {s_{46}\left\lbrack {c_{0} - e_{0}} \right\rbrack} + {s_{57}\left\lbrack {d_{0} - e_{0}} \right\rbrack}} \right)y} - e_{0}} = 0}} & (79)\end{matrix}$Again, the cubic polynomial equation (79) must be solved for y=X₁, andthen the solution can be substituted into (72)-(77) for the rest of thevariables. Actually, one only needs

$\begin{matrix}{\begin{matrix}{\left( \frac{P_{12}}{P_{02}} \right)_{11} = \left( \frac{\left\lbrack {TP}_{12}^{2} \right\rbrack + \left\lbrack {TP}_{12}^{1} \right\rbrack}{\left\lbrack {TP}_{02}^{2} \right\rbrack + \left\lbrack {TP}_{02}^{1} \right\rbrack} \right)_{11}} \\{= \frac{X_{4} + X_{6}}{X_{5} + X_{7}}} \\{= {\left( {\frac{c_{0}K_{1}^{4}y}{1 + {s_{46}y}} + \frac{c_{0}K_{1}^{6}y}{1 + {s_{46}y}}} \right)/\left( {\frac{d_{0}K_{1}^{5}y}{1 + {s_{57}y}} + \frac{d_{0}K_{1}^{7}y}{1 + {s_{57}y}}} \right)}}\end{matrix}{or}\text{}\begin{matrix}{\left( \frac{P_{12}}{P_{02}} \right)_{11} = {\left( \frac{c_{0}s_{46}y}{1 + {s_{46}y}} \right)/\left( \frac{d_{0}s_{57}y}{1 + {s_{57}y}} \right)}} \\{= {\frac{c_{0}}{d_{0}}\frac{s_{46}}{s_{57}}\frac{1 + {s_{57}y}}{1 + {s_{46}y}}}} \\{= {\frac{c_{0}}{d_{0}}\frac{s_{46}}{s_{57}}\frac{s_{57} + {1/y}}{s_{46} + {1/y}}}}\end{matrix}} & (80)\end{matrix}$where y solves (79).System Reduction—Full Model

As noted above, the system (52) of equations for the Full Model issimply the union of the systems (53) and (54) for models I and II,respectively, with the exception of the conservation rule for [T], i.e.,the equation for X₁. Therefore, while the equation for X₁ itself must behandled separately, the derivations from sections Model I and Model IIabove can be duplicated to obtain equations for all the variables interms of X₁. For convenience, one gathers the resulting equations in oneplace, as shown below.

$\begin{matrix}{{X_{2} = \frac{a_{0}K_{1}^{2}X_{1}}{1 + {X_{1}\left( {K_{1}^{2} + K_{1}^{8}} \right)}}}\mspace{14mu}\left( {{see}\mspace{14mu}(59)} \right)} & (81) \\{{X_{3} = \frac{b_{0}K_{1}^{3}X_{1}}{1 + {X_{1}\left( {K_{1}^{3} + K_{1}^{9}} \right)}}}\mspace{14mu}\left( {{see}\mspace{14mu}(60)} \right)} & (82) \\{{X_{4} = \frac{c_{0}K_{1}^{4}X_{1}}{1 + {X_{1}\left( {K_{1}^{4} + K_{1}^{6}} \right)}}}\mspace{14mu}\left( {{see}\mspace{14mu}(72)} \right)} & (83) \\{{X_{5} = \frac{d_{0}K_{1}^{5}X_{1}}{1 + {X_{1}\left( {K_{1}^{5} + K_{1}^{7}} \right)}}}\mspace{14mu}\left( {{see}\mspace{14mu}(73)} \right)} & (84) \\{{X_{6} = \frac{c_{0}K_{1}^{6}X_{1}}{1 + {X_{1}\left( {K_{1}^{4} + K_{1}^{6}} \right)}}}\mspace{14mu}\left( {{see}\mspace{14mu}(74)} \right)} & (85) \\{{X_{7} = \frac{d_{0}K_{1}^{7}X_{1}}{1 + {X_{1}\left( {K_{1}^{5} + K_{1}^{7}} \right)}}}\mspace{14mu}\left( {{see}\mspace{14mu}(75)} \right)} & (86) \\{{X_{8} = \frac{a_{0}K_{1}^{8}X_{1}}{1 + {X_{1}\left( {K_{1}^{2} + K_{1}^{8}} \right)}}}\mspace{14mu}\left( {{see}\mspace{14mu}(61)} \right)} & (87) \\{{X_{9} = \frac{b_{0}K_{1}^{9}X_{1}}{1 + {X_{1}\left( {K_{1}^{3} + K_{1}^{9}} \right)}}}\mspace{14mu}\left( {{see}\mspace{14mu}(62)} \right)} & (88) \\{{Y_{1} = \frac{a_{0}}{1 + {X_{1}\left( {K_{1}^{2} + K_{1}^{8}} \right)}}}\mspace{14mu}\left( {{see}\mspace{14mu}(63)} \right)} & (89) \\{{Y_{2} = \frac{b_{0}}{1 + {X_{1}\left( {K_{1}^{3} + K_{1}^{9}} \right)}}}\mspace{14mu}\left( {{see}\mspace{14mu}(64)} \right)} & (90) \\{{Y_{3} = \frac{c_{0}}{1 + {X_{1}\left( {K_{1}^{4} + K_{1}^{6}} \right)}}}\mspace{14mu}\left( {{see}\mspace{14mu}(76)} \right)} & (91) \\{{Y_{4} = \frac{d_{0}}{1 + {X_{1}\left( {K_{1}^{5} + K_{1}^{7}} \right)}}}\mspace{14mu}\left( {{see}\mspace{14mu}(77)} \right)} & (92)\end{matrix}$It remains to derive the univariate equation in X₁. Since the terms (K₁²+K₁ ⁸), (K₁ ³+K₁ ⁹), (K₁ ⁴+K₁ ⁶), and (K₁ ⁵+K₁ ⁷) appear frequently inthe following derivation, as in the previous sections, we abbreviatethese terms with the short-hand notation given below. As in discussionsof Models I and II above, let

$\begin{matrix}{{s_{28} \equiv {K_{1}^{2} + K_{1}^{8}}};} & {{s_{39} \equiv {K_{1}^{3} + K_{1}^{9}}};} \\{{s_{46} \equiv {K_{1}^{4} + K_{1}^{6}}};} & {{s_{57} \equiv {K_{1}^{5} + K_{1}^{7}}};}\end{matrix}$ and  let  z ≡ X₁.Note again that a different symbol for X₁ has to be employed to avoidconfusion with equations (66) and (79).

$\begin{matrix}{\begin{matrix}{e_{0} = {X_{1} + X_{2} + X_{3} + X_{4} + X_{5} + X_{6} + X_{7} + X_{8} + {X_{9}\mspace{14mu}\left( {{by}\mspace{14mu}(52)} \right)}}} \\{= {z + {z\;\frac{a_{0}K_{1}^{2}}{1 + {s_{28}z}}} + {z\frac{b_{0}K_{1}^{3}}{1 + {s_{39}z}}} + {z\;\frac{c_{0}K_{1}^{4}}{1 + {s_{46}z}}} +}} \\{{z\;\frac{d_{0}K_{1}^{5}}{1 + {s_{57}z}}\mspace{14mu}\left( {{{by}\mspace{14mu}(81)} - (84)} \right)} +} \\{{z\frac{\;{c_{0}K_{1}^{6}}}{1 + {s_{46}z}}} + {z\frac{\;{d_{0}K_{1}^{7}}}{1 + {s_{57}z}}} + {z\;\frac{a_{0}K_{1}^{8}}{1 + {zs}_{28}}} + {z\;\frac{b_{0}K_{1}^{9}}{1 + {zs}_{39}}\mspace{14mu}\left( {{{by}\mspace{14mu}(85)} - (88)} \right)}} \\{= {z\left\lbrack {1 + {a_{0}\frac{K_{1}^{2} + K_{1}^{8}}{1 + {s_{28}z}}} + {b_{0}\frac{K_{1}^{3} + K_{1}^{9}}{1 + {s_{39}z}}} + {c_{0}\frac{K_{1}^{4} + K_{1}^{6}}{1 + {s_{46}z}}} + {d_{0}\frac{K_{1}^{5} + K_{1}^{7}}{1 + {s_{57}z}}}} \right\rbrack}}\end{matrix}{or}} & (93) \\{{e_{0} = {z\left\lbrack {1 + \frac{a_{0}s_{28}}{1 + {s_{28}z}} + \frac{b_{0}s_{39}}{1 + {s_{39}z}} + \frac{c_{0}s_{46}}{1 + {s_{46}z}} + \frac{d_{0}s_{57}}{1 + {s_{57}z}}} \right\rbrack}}{or}} & (94) \\\begin{matrix}{{\begin{matrix}{\left( {1 + {s_{28}z}} \right)\left( {1 + {s_{39}z}} \right)} \\{\left( {1 + {s_{46}z}} \right)\left( {1 + {s_{57}z}} \right)}\end{matrix}e_{0}} = {z\left\lbrack {\left( {1 + {s_{28}z}} \right)\left( {1 + {s_{39}z}} \right)\left( {1 + {s_{46}z}} \right)} \right.}} \\{\left( {1 + {s_{57}z}} \right) + {a_{0}{s_{28}\left( {1 + {s_{39}z}} \right)}\left( {1 + {s_{46}z}} \right)}} \\{\left( {1 + {s_{57}z}} \right) + {b_{0}{s_{39}\left( {1 + {s_{28}z}} \right)}\left( {1 + {s_{46}z}} \right)}} \\{\left( {1 + {s_{57}z}} \right) + {c_{0}{s_{46}\left( {1 + {s_{28}z}} \right)}\left( {1 + {s_{39}z}} \right)}} \\{\left( {1 + {s_{57}z}} \right) + {d_{0}{s_{57}\left( {1 + {s_{28}z}} \right)}\left( {1 + {s_{39}z}} \right)}} \\\left. \left( {1 + {s_{46}z}} \right) \right\rbrack\end{matrix} & (95)\end{matrix}$Since one now has a 5_(th) order polynomial equation in z to solve, andsince its roots cannot be expressed symbolically in a closed form, onemust resort to a purely numerical approach. Nonetheless, thematch-to-mismatch ratio signals can be obtained in terms of z.

$\begin{matrix}{\begin{matrix}{\left( \frac{P_{11}}{P_{01}} \right)_{full} = \left( \frac{\left\lbrack {TP}_{11}^{1} \right\rbrack + \left\lbrack {TP}_{01}^{2} \right\rbrack}{\left\lbrack {TP}_{01}^{1} \right\rbrack + \left\lbrack {TP}_{01}^{2} \right\rbrack} \right)_{full}} \\{= \frac{X_{2} + X_{8}}{X_{3} + X_{9}}} \\{= {\frac{a_{0}}{b_{0}}\frac{s_{28}}{s_{39}}\frac{1 + {s_{39}z}}{1 + {s_{28}z}}}} \\{= {\frac{a_{0}}{b_{0}}\frac{s_{28}}{s_{39}}\frac{s_{39} + {1/z}}{s_{28} + {1/z}}\mspace{14mu}\left( {{see}\mspace{14mu}\left( (67) \right)} \right.}}\end{matrix}{and}} & (96) \\{\begin{matrix}{\left( \frac{P_{12}}{P_{02}} \right)_{full} = \left( \frac{\left\lbrack {TP}_{12}^{2} \right\rbrack + \left\lbrack {TP}_{12}^{1} \right\rbrack}{\left\lbrack {TP}_{02}^{2} \right\rbrack + \left\lbrack {TP}_{02}^{1} \right\rbrack} \right)_{full}} \\{= \frac{X_{4} + X_{6}}{X_{5} + X_{7}}} \\{= {\frac{c_{0}}{d_{0}}\frac{s_{46}}{s_{57}}\frac{1 + {s_{57}z}}{1 + {s_{46}z}}}} \\{{= {\frac{c_{0}}{d_{0}}\frac{s_{46}}{s_{57}}\frac{s_{57} + {1/z}}{s_{46} + {1/z}}\mspace{14mu}\left( {{see}\mspace{14mu}(80)} \right)}},}\end{matrix}{{where}\mspace{14mu} z\mspace{14mu}{solves}\mspace{14mu}{(95).}}} & (97)\end{matrix}$Normalized Discrimination: Δ-Plot

To display the results of the computation and to describe the principaleffects of competitive hybridization in a graphical manner, a“normalized discrimination” plot is provided, also referred to herein asa Δ-plot. This plot displays, in a manner that is independent of thespecific sequences of a PM/MM probe pair, the normalized discrimination,herein denoted by Δ, and denoted in FIG. 4 by the ratio “pm/mm”, as afunction of the molar ratio, [T]₀/Σ[P]₀ of the initial targetconcentration and the sum of initial probe concentrations. Here, thenormalized discrimination represents the ratio of the amounts ofprobe-target complexes formed by PM and MM probes, normalized by therespective sequence-specific affinities. To the extent that affinities,calculated by summing over sequence-specific NN interactions on theentire region of the interactions, stabilize a duplex, the shape of thenormalized discrimination plot is independent of the specific probesequences under consideration. This makes the Δ-plot a valuable tool forthe determination of the effect of competition wherein competitivehybridization manifests itself in the form of a shift of the Δ-plot fora single PM/MM probe pair in the presence of a second (or additional)PM/MM pair(s).

Discrimination is lowest in the presence of excess amounts of targetbecause even the MM probe, while interacting more weakly with the targetthan the PM probe, will capture large amounts of target and generate alarge signal. Conversely, discrimination is largest in thetarget-depleted regime: in the extreme case, a single target moleculewould have to select the PM probe over the MM probe, producing infinitediscrimination but at the expense of a very weak signal. Preferably,multiplexed analysis is thus carried out under conditions of slighttarget depletion so as to maximize discrimination while retaining anacceptable signal intensity to facilitate experimental measurements.

FIGS. 4A and 4B display the effects of competitive hybridization for apair of PM/MM probes forming a complex with the same target, here Exon11 of the CFTR gene, in accordance with the model of FIG. 2. FIG. 4Ashows a shift “UP” of the Δ-plot of the first PM/MM pair (Model I) inthe presence of the second PM/MM probe pair (Full Model), while FIG. 4Bshows a shift “DOWN” of the Δ-plot of the second PM/MM pair (Model II)in the presence of the first PM/MM probe pair (Full Model). These Δ-plotshifts generally will have a dramatic effect upon the discrimination, A,attainable for a given initial molar ratio of target to probes. That is,the discrimination between perfect match and mismatch attained in anexperimental design involving a single PM/MM probe pair for a giventarget can either improve, as in the case depicted in FIG. 4A, ordeteriorate, as in the case depicted in FIG. 4B, with the addition of asecond PM/MM pair directed to the same target.

Predicting the Effects of Competitive Hybridization

The methods described herein permit the prediction of the shifts in theΔ-plot, given the sequences of two or more PM/MM probe pairs. Considertwo probes, each having associated with it the pair {P_(.,pm),P_(.,mm)}. For each probe, the pm/mm ratio shifts up or down in thepresence of the other probe. The direction of the shift was determinedto be a function of the relative sizes of the affinity constants K,where cross-bound states can be neglected. For a given probe, letK_(T)pm, K_(T)mm denote the affinity constants for this probe's bindingsite with pm and mm, respectively; let K_(O)pm, K_(O)mm be the otherprobe's affinity constants with pm and mm. View the competition effectas a binary function on the space of affinity constants (+1 for up, −1for down shift) and consider the projection of the affinity constantspace R⁴={K_(T)pm, K_(T)mm, K_(O)pm, K_(O)mm} onto the plane L with axeslog(K_(T)pm/K_(O)pm) and log(K_(T)mm/K_(O)mm). On this plane, thecompetition effect function values can be clearly separated by the linex+y=0. This condition holds for physical exon 11 CFTR probes, as shownin FIGS. 5A and 5B.

The empirically determined condition can be described by the followinglogically equivalent statements:

-   -   pm/mm ratio shifts up        y<−χ        log(K _(T) mm/K _(O) mm)<−log(K _(T) pm/K _(O) pm)        K_(T)pmK_(T)mm<K_(O)pmK_(O)mm  (133)        Refer to the “empirical design rule” for comparison of a first        pair of PM/MM probes (1) and second pair of PM/MM probes (2):    -   Δ(PM/MM Pair 1) increases in presence of PM/MM Pair 2        -   whenever            K₁ ^(PM)K₁ ^(MM)<K₂ ^(PM)K₂ ^(MM)            Design of Optimal Probe Pairs for Competitive Hybridization

The reliability in predicting Δ-plot shifts renders the empirical designrule, derived herein from the heuristic analysis described above, avaluable design tool to guide the selection of PM/MM probe pairs formultiplexed hybridization analysis under conditions permittingcompetitive hybridization. The predictive power can be applied toexperimental designs as follows.

Suppose an application (e.g., HLA typing) requires the use of several(n) probes. Let the probes be ranked in order of importance to theproblem: A, B, C, D, E, F . . . . One can choose the alternates forprobes such that the ratios for important probes are improved. Recallthat according to equation (133), the pm/mm ratio for this probeincreases wheneverK_(T)pmK_(T)mm<K_(O)pmK_(O)mmThe process of selecting probes and alternates can be described indetail as follows. The DNA bases in an oligonucleotide are denoted byΣ={A, C, G, T}.Start with probe A (most important); consider adding probe B. Theoptions are:

-   -   Use probe B with alternate B′ differing in:        -   Position 1, value ΕΣ−B[1];        -   Position 2, value εΣ−B[2];        -   . . .        -   Position N, value εΣ−B[N];    -   Do not use probe B.        (Note that, here, This probe=A; Other probe=B.)        Construction of Probe Pairs

Consider the three capture probes from the probe selection exampleabove, designed for the analysis of mutations in exon 11 of the cysticfibrosis transmembrane regulator (CFTR) gene, where A=C381, B=A327,C=D359.

Pairwise computational analysis using the methods described herein aboveindicates that:

-   -   A327 will improve A(C381), the discrimination for C381;    -   D359 will improve A(A327), the discrimination for A327.        The empirical design rule herein identified indicates that it        therefore follows that:    -   D359 will improve A(C381), the discrimination for C381.        This prediction was tested by invoking the “extended model” for        three probe pairs as described herein and comparing the        resulting Δ-plots for the following cases (FIG. 7):

Δ(C381) in presence of both right-most Δ-plot in FIG. 7 A327 and D359Δ(C381) in presence of middle Δ-plot in FIG. 7 A327 only Δ(C381) onlyleft-most Δ-plot in FIG. 7For a given initial target concentration, for example, corresponding toa value of 0.7 of the abscissa variable, Δ(C381) will increase uponaddition of A327 and will increase further upon addition of D359.Experimental Illustration of the Effects of Competitive Hybridization

FIGS. 8A to 8D illustrate another potential application of the concept:introducing spurious “booster” probes into the multiplexed reaction forthe sole purpose of improving the signal for probes of interest. FIG. 8Ashows the results of hybridization of capture probes C381 to regions ofthe exon 11 of the cystic fibrosis transmembrane regulator (CFTR) genein the presence of capture probes referred to as A327 and D359 (asdepicted in FIG. 8B), when PM/MM probe pairs are present for A327 andC381, but only the PM probe is present for D359 (the “booster” probe inthis example). The effect on the signal of probe C381 (as depicted inFIG. 8A) is analogous to that shown above (in FIG. 7): addition of probeA327 improves the signal for probe C381 (from “Extended Partial Model:1” to “Pairwise Model: 1&2”), and further addition of “booster” probeD359 improves the signal for C381 even more (to “Extended Full Model”).The trade-off inherent in the empirical design rule shows that thesignal for probe A327 (depicted in FIG. 8C, “Extended Partial Model: 2”)deteriorates with the addition of probe C381 (“Pairwise Model: 1&2”).However, addition of “booster” probe D359 compensates for thedeterioration, resulting in the improvement of the signal for probe A327in the presence of both C381 and D359 (“Extended Full Model”). FIG. 8Dshows that the “booster” probe D359 has no discrimination signal,regardless of which other probes it is multiplexed with, as expected,since no MM probe was included for probe D359. Nonetheless, since probeD359 is introduced as a spurious probe (i.e., its signal was never ofinterest), it fully serves its purpose as a “booster” probe by improvingthe signal of preceding probes.

This example illustrates that adding probes which bind competitively tothe target can increase the signal generated if subsequent probes arechosen appropriately, demonstrating that the empirical design rule isborne out.

3—Probe Pooling

In some applications, such as mutation analysis (e.g., for CF), one doesnot have the freedom of choosing the sequence of the mismatchprobe—“mutant” matching sequence is used as “wild-type” mismatch andvice versa. In those cases, the options listed above relating toselection of probes [appearing below the formula K_(T) ^(PM)K_(T)^(MM)<K_(O) ^(PM)K_(O) ^(MM)] are reduced to:

-   -   Use probe B with the given alternate (dictated by “mutant”        sequence), or    -   Do not use probe B.

Furthermore, there are applications where one cannot rank probes byimportance—all probes are equally important. Hence, discarding a probeis not really an option. For these applications, one can easily extendthe approach for selecting probes and alternates, described above, intoa method for pooling probes, namely, sorting through equally importantprobes and separating them into groups of non-interfering probes.Starting with a randomly ordered list of n equally important probes, wego through the probe selection process (above), with the exception thatwhenever our method dictates that we discard a given probe, we use thatprobe to start a new “pool”. Each subsequent probe gets placed bystarting with the initial probe pool, and in the event it is to be“discarded” from it, cycles to the next available “pool”. If allcurrently existing probe pools recommend that it be discarded, we use itto start a new “pool”. In this manner, the original list of n probesgets separated into k groups, k<n.

One may also choose to adjust the selection criteria for a newly addedprobe to not only boost the signals of the existing probes in the pool,but also limit the decay of the signal for the newly added probe itself.

Experimental Results

FIGS. 9A-9E, 10A, and 10B show experimental results of combining probesand targets as shown in the figures, and as listed in the table in FIG.9A. In FIG. 10B, adding the probe with the higher affinity (D359)together with the A327 probe to the target caused an increase indiscrimination (panel C) as predicted by the empirical design rule. InFIG. 10A, adding the lower affinity probe (C381) together with the A327probe to the target caused an increase in discrimination (panel C),which is opposite to the prediction of the empirical design rule. Theresults of various probe combinations in FIGS. 9A-9E are also notconsistent with the empirical design rule. These inconsistenciesindicate that the nearest neighbor model of affinity (used to calculatethe affinity constants shown in the table in FIG. 6B) has limitations.

It should be understood that the examples, terms, expressions, andmethods described herein are exemplary only, and not limiting, and thatthe invention is defined only in the claims which follow, and includesall equivalents of the subject matter of those claims.

1. A method of selecting a set of oligonucleotide probe pairs for use ina hybridization assay to determine the sequences at particular sites ina target sequence, wherein one member of a particular pair has adifferent sequence than the other member and wherein one member (the “PMprobe”) has a PM subsequence perfectly complementary to a normal,non-variant subsequence of the target sequence and the other memberprobe of that pair (the “MM probe”) differs in sequence at a number ofnucleotide positions from said one member and has an MM subsequencecomplementary to a known variant of said subsequence, and whereindifferent probe pairs have members including PM and/or MM subsequenceswhich are complementary to different subsequences of the same target,and wherein the selection method is designed to maximize discrimination(where discrimination is determined from the ratio of the intensity ofthe signals generated from labels attached to the probes, a greaterintensity of a signal of particular member probes indicating more ofsaid particular member probes being annealed to said target) betweenmembers of a particular probe pair, when said target is annealed to twoor more of said different probe pairs, the method comprising: a)selecting a number of candidate probe pairs; b) adding a first probepair from said number of candidate probe pairs to the set; c) addingadditional probe pairs to the set if, for each successive probe pair tobe added, said probe pair satisfies the inequality:K_(T) ^(PM)K_(T) ^(MM)<K_(O) ^(PM)K_(O) ^(MM), where: K_(T) ^(PM) is theaffinity constant for the PM probe in the immediately-preceding-addedprobe pair binding to its designated target subsequence; K_(T) ^(MM) isthe affinity constant for the MM probe in theimmediately-preceding-added probe pair binding to its designated targetsubsequence; K_(o) ^(PM) is the affinity constant for the PM probe inthe next successive probe pair binding to its designated targetsubsequence; and K_(O) ^(MM) is the affinity constant for the MM probein the next successive probe pair binding to its designated targetsubsequence; and d) using the set of probe pairs obtained from step (c)in a hybridization assay to determine which of said target subsequencesare normal and which are variant.
 2. The method of claim 1 wherein theMM probe of the first and the additional probe pairs differ in sequencefrom the PM probe of each of said pairs by one nucleotide.
 3. The methodof claim 1 wherein the probe pairs are attached to encoded beads,wherein the encoding indicates the sequence of the attached probe.